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Complete Divisor Transforms, Consistent Mobius Indexing and Roundtrips

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batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations · 2026-10-14 · 100 questions

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1. The Forward Transform Sums over All Positive Divisors

Use positive integer n=1, exact integer arithmetic function f(k)=a*k+b with a=1, b=-12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(1, [[1, -11]], -11, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (1, [[1, -11]], -11, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

2. The Forward Transform Sums over All Positive Divisors

Use positive integer n=2, exact integer arithmetic function f(k)=a*k+b with a=2, b=-11. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(2, [[1, -9], [2, -7]], -16, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (2, [[1, -9], [2, -7]], -16, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

3. The Forward Transform Sums over All Positive Divisors

Use positive integer n=3, exact integer arithmetic function f(k)=a*k+b with a=3, b=-10. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(3, [[1, -7], [3, -1]], -8, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (3, [[1, -7], [3, -1]], -8, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

4. The Forward Transform Sums over All Positive Divisors

Use positive integer n=4, exact integer arithmetic function f(k)=a*k+b with a=4, b=-9. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(4, [[1, -5], [2, -1], [4, 7]], 1, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (4, [[1, -5], [2, -1], [4, 7]], 1, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

5. The Forward Transform Sums over All Positive Divisors

Use positive integer n=5, exact integer arithmetic function f(k)=a*k+b with a=5, b=-8. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(5, [[1, -3], [5, 17]], 14, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (5, [[1, -3], [5, 17]], 14, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

6. The Forward Transform Sums over All Positive Divisors

Use positive integer n=6, exact integer arithmetic function f(k)=a*k+b with a=1, b=-7. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(6, [[1, -6], [2, -5], [3, -4], [6, -1]], -16, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (6, [[1, -6], [2, -5], [3, -4], [6, -1]], -16, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

7. The Forward Transform Sums over All Positive Divisors

Use positive integer n=7, exact integer arithmetic function f(k)=a*k+b with a=2, b=-6. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(7, [[1, -4], [7, 8]], 4, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (7, [[1, -4], [7, 8]], 4, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

8. The Forward Transform Sums over All Positive Divisors

Use positive integer n=8, exact integer arithmetic function f(k)=a*k+b with a=3, b=-5. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(8, [[1, -2], [2, 1], [4, 7], [8, 19]], 25, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (8, [[1, -2], [2, 1], [4, 7], [8, 19]], 25, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

9. The Forward Transform Sums over All Positive Divisors

Use positive integer n=9, exact integer arithmetic function f(k)=a*k+b with a=4, b=-4. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(9, [[1, 0], [3, 8], [9, 32]], 40, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (9, [[1, 0], [3, 8], [9, 32]], 40, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

10. The Forward Transform Sums over All Positive Divisors

Use positive integer n=10, exact integer arithmetic function f(k)=a*k+b with a=5, b=-3. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(10, [[1, 2], [2, 7], [5, 22], [10, 47]], 78, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (10, [[1, 2], [2, 7], [5, 22], [10, 47]], 78, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

11. The Forward Transform Sums over All Positive Divisors

Use positive integer n=11, exact integer arithmetic function f(k)=a*k+b with a=1, b=-2. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(11, [[1, -1], [11, 9]], 8, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (11, [[1, -1], [11, 9]], 8, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

12. The Forward Transform Sums over All Positive Divisors

Use positive integer n=12, exact integer arithmetic function f(k)=a*k+b with a=2, b=-1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(12, [[1, 1], [2, 3], [3, 5], [4, 7], [6, 11], [12, 23]], 50, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (12, [[1, 1], [2, 3], [3, 5], [4, 7], [6, 11], [12, 23]], 50, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

13. The Forward Transform Sums over All Positive Divisors

Use positive integer n=13, exact integer arithmetic function f(k)=a*k+b with a=3, b=0. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(13, [[1, 3], [13, 39]], 42, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (13, [[1, 3], [13, 39]], 42, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

14. The Forward Transform Sums over All Positive Divisors

Use positive integer n=14, exact integer arithmetic function f(k)=a*k+b with a=4, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(14, [[1, 5], [2, 9], [7, 29], [14, 57]], 100, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (14, [[1, 5], [2, 9], [7, 29], [14, 57]], 100, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

15. The Forward Transform Sums over All Positive Divisors

Use positive integer n=15, exact integer arithmetic function f(k)=a*k+b with a=5, b=2. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(15, [[1, 7], [3, 17], [5, 27], [15, 77]], 128, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (15, [[1, 7], [3, 17], [5, 27], [15, 77]], 128, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

16. The Forward Transform Sums over All Positive Divisors

Use positive integer n=16, exact integer arithmetic function f(k)=a*k+b with a=1, b=3. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(16, [[1, 4], [2, 5], [4, 7], [8, 11], [16, 19]], 46, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (16, [[1, 4], [2, 5], [4, 7], [8, 11], [16, 19]], 46, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

17. The Forward Transform Sums over All Positive Divisors

Use positive integer n=17, exact integer arithmetic function f(k)=a*k+b with a=2, b=4. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(17, [[1, 6], [17, 38]], 44, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (17, [[1, 6], [17, 38]], 44, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

18. The Forward Transform Sums over All Positive Divisors

Use positive integer n=18, exact integer arithmetic function f(k)=a*k+b with a=3, b=5. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(18, [[1, 8], [2, 11], [3, 14], [6, 23], [9, 32], [18, 59]], 147, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (18, [[1, 8], [2, 11], [3, 14], [6, 23], [9, 32], [18, 59]], 147, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

19. The Forward Transform Sums over All Positive Divisors

Use positive integer n=19, exact integer arithmetic function f(k)=a*k+b with a=4, b=6. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(19, [[1, 10], [19, 82]], 92, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (19, [[1, 10], [19, 82]], 92, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

20. The Forward Transform Sums over All Positive Divisors

Use positive integer n=20, exact integer arithmetic function f(k)=a*k+b with a=5, b=7. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(20, [[1, 12], [2, 17], [4, 27], [5, 32], [10, 57], [20, 107]], 252, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (20, [[1, 12], [2, 17], [4, 27], [5, 32], [10, 57], [20, 107]], 252, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

21. The Forward Transform Sums over All Positive Divisors

Use positive integer n=21, exact integer arithmetic function f(k)=a*k+b with a=1, b=8. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(21, [[1, 9], [3, 11], [7, 15], [21, 29]], 64, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (21, [[1, 9], [3, 11], [7, 15], [21, 29]], 64, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

22. The Forward Transform Sums over All Positive Divisors

Use positive integer n=22, exact integer arithmetic function f(k)=a*k+b with a=2, b=9. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(22, [[1, 11], [2, 13], [11, 31], [22, 53]], 108, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (22, [[1, 11], [2, 13], [11, 31], [22, 53]], 108, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

23. The Forward Transform Sums over All Positive Divisors

Use positive integer n=23, exact integer arithmetic function f(k)=a*k+b with a=3, b=10. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(23, [[1, 13], [23, 79]], 92, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (23, [[1, 13], [23, 79]], 92, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

24. The Forward Transform Sums over All Positive Divisors

Use positive integer n=24, exact integer arithmetic function f(k)=a*k+b with a=4, b=11. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(24, [[1, 15], [2, 19], [3, 23], [4, 27], [6, 35], [8, 43], [12, 59], [24, 107]], 328, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (24, [[1, 15], [2, 19], [3, 23], [4, 27], [6, 35], [8, 43], [12, 59], [24, 107]], 328, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

25. The Forward Transform Sums over All Positive Divisors

Use positive integer n=25, exact integer arithmetic function f(k)=a*k+b with a=5, b=12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).

View answer and explanation

(25, [[1, 17], [5, 37], [25, 137]], 191, sum_over_all_positive_divisors)

List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (25, [[1, 17], [5, 37], [25, 137]], 191, sum_over_all_positive_divisors).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-forward-divisor-transform · candidate

26. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=1, exact integer arithmetic function f(k)=a*k+b with a=1, b=-12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(1, [[1, 1, 1, -11, -11]], -11, -11, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (1, [[1, 1, 1, -11, -11]], -11, -11, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

27. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=2, exact integer arithmetic function f(k)=a*k+b with a=2, b=-11. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(2, [[1, 1, 2, -16, -16], [2, -1, 1, -9, 9]], -7, -7, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (2, [[1, 1, 2, -16, -16], [2, -1, 1, -9, 9]], -7, -7, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

28. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=3, exact integer arithmetic function f(k)=a*k+b with a=3, b=-10. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(3, [[1, 1, 3, -8, -8], [3, -1, 1, -7, 7]], -1, -1, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (3, [[1, 1, 3, -8, -8], [3, -1, 1, -7, 7]], -1, -1, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

29. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=4, exact integer arithmetic function f(k)=a*k+b with a=4, b=-9. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(4, [[1, 1, 4, 1, 1], [2, -1, 2, -6, 6], [4, 0, 1, -5, 0]], 7, 7, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (4, [[1, 1, 4, 1, 1], [2, -1, 2, -6, 6], [4, 0, 1, -5, 0]], 7, 7, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

30. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=5, exact integer arithmetic function f(k)=a*k+b with a=5, b=-8. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(5, [[1, 1, 5, 14, 14], [5, -1, 1, -3, 3]], 17, 17, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (5, [[1, 1, 5, 14, 14], [5, -1, 1, -3, 3]], 17, 17, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

31. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=6, exact integer arithmetic function f(k)=a*k+b with a=1, b=-7. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(6, [[1, 1, 6, -16, -16], [2, -1, 3, -10, 10], [3, -1, 2, -11, 11], [6, 1, 1, -6, -6]], -1, -1, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (6, [[1, 1, 6, -16, -16], [2, -1, 3, -10, 10], [3, -1, 2, -11, 11], [6, 1, 1, -6, -6]], -1, -1, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

32. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=7, exact integer arithmetic function f(k)=a*k+b with a=2, b=-6. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(7, [[1, 1, 7, 4, 4], [7, -1, 1, -4, 4]], 8, 8, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (7, [[1, 1, 7, 4, 4], [7, -1, 1, -4, 4]], 8, 8, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

33. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=8, exact integer arithmetic function f(k)=a*k+b with a=3, b=-5. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(8, [[1, 1, 8, 25, 25], [2, -1, 4, 6, -6], [4, 0, 2, -1, 0], [8, 0, 1, -2, 0]], 19, 19, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (8, [[1, 1, 8, 25, 25], [2, -1, 4, 6, -6], [4, 0, 2, -1, 0], [8, 0, 1, -2, 0]], 19, 19, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

34. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=9, exact integer arithmetic function f(k)=a*k+b with a=4, b=-4. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(9, [[1, 1, 9, 40, 40], [3, -1, 3, 8, -8], [9, 0, 1, 0, 0]], 32, 32, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (9, [[1, 1, 9, 40, 40], [3, -1, 3, 8, -8], [9, 0, 1, 0, 0]], 32, 32, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

35. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=10, exact integer arithmetic function f(k)=a*k+b with a=5, b=-3. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(10, [[1, 1, 10, 78, 78], [2, -1, 5, 24, -24], [5, -1, 2, 9, -9], [10, 1, 1, 2, 2]], 47, 47, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (10, [[1, 1, 10, 78, 78], [2, -1, 5, 24, -24], [5, -1, 2, 9, -9], [10, 1, 1, 2, 2]], 47, 47, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

36. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=11, exact integer arithmetic function f(k)=a*k+b with a=1, b=-2. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(11, [[1, 1, 11, 8, 8], [11, -1, 1, -1, 1]], 9, 9, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (11, [[1, 1, 11, 8, 8], [11, -1, 1, -1, 1]], 9, 9, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

37. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=12, exact integer arithmetic function f(k)=a*k+b with a=2, b=-1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(12, [[1, 1, 12, 50, 50], [2, -1, 6, 20, -20], [3, -1, 4, 11, -11], [4, 0, 3, 6, 0], [6, 1, 2, 4, 4], [12, 0, 1, 1, 0]], 23, 23, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (12, [[1, 1, 12, 50, 50], [2, -1, 6, 20, -20], [3, -1, 4, 11, -11], [4, 0, 3, 6, 0], [6, 1, 2, 4, 4], [12, 0, 1, 1, 0]], 23, 23, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

38. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=13, exact integer arithmetic function f(k)=a*k+b with a=3, b=0. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(13, [[1, 1, 13, 42, 42], [13, -1, 1, 3, -3]], 39, 39, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (13, [[1, 1, 13, 42, 42], [13, -1, 1, 3, -3]], 39, 39, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

39. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=14, exact integer arithmetic function f(k)=a*k+b with a=4, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(14, [[1, 1, 14, 100, 100], [2, -1, 7, 34, -34], [7, -1, 2, 14, -14], [14, 1, 1, 5, 5]], 57, 57, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (14, [[1, 1, 14, 100, 100], [2, -1, 7, 34, -34], [7, -1, 2, 14, -14], [14, 1, 1, 5, 5]], 57, 57, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

40. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=15, exact integer arithmetic function f(k)=a*k+b with a=5, b=2. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(15, [[1, 1, 15, 128, 128], [3, -1, 5, 34, -34], [5, -1, 3, 24, -24], [15, 1, 1, 7, 7]], 77, 77, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (15, [[1, 1, 15, 128, 128], [3, -1, 5, 34, -34], [5, -1, 3, 24, -24], [15, 1, 1, 7, 7]], 77, 77, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

41. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=16, exact integer arithmetic function f(k)=a*k+b with a=1, b=3. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(16, [[1, 1, 16, 46, 46], [2, -1, 8, 27, -27], [4, 0, 4, 16, 0], [8, 0, 2, 9, 0], [16, 0, 1, 4, 0]], 19, 19, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (16, [[1, 1, 16, 46, 46], [2, -1, 8, 27, -27], [4, 0, 4, 16, 0], [8, 0, 2, 9, 0], [16, 0, 1, 4, 0]], 19, 19, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

42. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=17, exact integer arithmetic function f(k)=a*k+b with a=2, b=4. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(17, [[1, 1, 17, 44, 44], [17, -1, 1, 6, -6]], 38, 38, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (17, [[1, 1, 17, 44, 44], [17, -1, 1, 6, -6]], 38, 38, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

43. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=18, exact integer arithmetic function f(k)=a*k+b with a=3, b=5. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(18, [[1, 1, 18, 147, 147], [2, -1, 9, 54, -54], [3, -1, 6, 56, -56], [6, 1, 3, 22, 22], [9, 0, 2, 19, 0], [18, 0, 1, 8, 0]], 59, 59, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (18, [[1, 1, 18, 147, 147], [2, -1, 9, 54, -54], [3, -1, 6, 56, -56], [6, 1, 3, 22, 22], [9, 0, 2, 19, 0], [18, 0, 1, 8, 0]], 59, 59, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

44. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=19, exact integer arithmetic function f(k)=a*k+b with a=4, b=6. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(19, [[1, 1, 19, 92, 92], [19, -1, 1, 10, -10]], 82, 82, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (19, [[1, 1, 19, 92, 92], [19, -1, 1, 10, -10]], 82, 82, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

45. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=20, exact integer arithmetic function f(k)=a*k+b with a=5, b=7. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(20, [[1, 1, 20, 252, 252], [2, -1, 10, 118, -118], [4, 0, 5, 44, 0], [5, -1, 4, 56, -56], [10, 1, 2, 29, 29], [20, 0, 1, 12, 0]], 107, 107, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (20, [[1, 1, 20, 252, 252], [2, -1, 10, 118, -118], [4, 0, 5, 44, 0], [5, -1, 4, 56, -56], [10, 1, 2, 29, 29], [20, 0, 1, 12, 0]], 107, 107, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

46. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=21, exact integer arithmetic function f(k)=a*k+b with a=1, b=8. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(21, [[1, 1, 21, 64, 64], [3, -1, 7, 24, -24], [7, -1, 3, 20, -20], [21, 1, 1, 9, 9]], 29, 29, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (21, [[1, 1, 21, 64, 64], [3, -1, 7, 24, -24], [7, -1, 3, 20, -20], [21, 1, 1, 9, 9]], 29, 29, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

47. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=22, exact integer arithmetic function f(k)=a*k+b with a=2, b=9. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(22, [[1, 1, 22, 108, 108], [2, -1, 11, 42, -42], [11, -1, 2, 24, -24], [22, 1, 1, 11, 11]], 53, 53, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (22, [[1, 1, 22, 108, 108], [2, -1, 11, 42, -42], [11, -1, 2, 24, -24], [22, 1, 1, 11, 11]], 53, 53, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

48. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=23, exact integer arithmetic function f(k)=a*k+b with a=3, b=10. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(23, [[1, 1, 23, 92, 92], [23, -1, 1, 13, -13]], 79, 79, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (23, [[1, 1, 23, 92, 92], [23, -1, 1, 13, -13]], 79, 79, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

49. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=24, exact integer arithmetic function f(k)=a*k+b with a=4, b=11. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(24, [[1, 1, 24, 328, 328], [2, -1, 12, 178, -178], [3, -1, 8, 104, -104], [4, 0, 6, 92, 0], [6, 1, 4, 61, 61], [8, 0, 3, 38, 0], [12, 0, 2, 34, 0], [24, 0, 1, 15, 0]], 107, 107, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (24, [[1, 1, 24, 328, 328], [2, -1, 12, 178, -178], [3, -1, 8, 104, -104], [4, 0, 6, 92, 0], [6, 1, 4, 61, 61], [8, 0, 3, 38, 0], [12, 0, 2, 34, 0], [24, 0, 1, 15, 0]], 107, 107, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

50. Keep mu(d) Paired with F(n Divided by d)

Use positive integer n=25, exact integer arithmetic function f(k)=a*k+b with a=5, b=12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

View answer and explanation

(25, [[1, 1, 25, 191, 191], [5, -1, 5, 54, -54], [25, 0, 1, 17, 0]], 137, 137, Yes)

Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (25, [[1, 1, 25, 191, 191], [5, -1, 5, 54, -54], [25, 0, 1, 17, 0]], 137, 137, Yes).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-inverse-reconstruction · candidate

51. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=1, exact integer arithmetic function f(k)=a*k+b with a=1, b=-12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(1, [[1, -11, -11, -11]], Yes, mu_one_boundary_preserved)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (1, [[1, -11, -11, -11]], Yes, mu_one_boundary_preserved).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

52. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=2, exact integer arithmetic function f(k)=a*k+b with a=2, b=-11. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(2, [[1, -9, -9, -9], [2, -16, -7, -7]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (2, [[1, -9, -9, -9], [2, -16, -7, -7]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

53. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=3, exact integer arithmetic function f(k)=a*k+b with a=3, b=-10. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(3, [[1, -7, -7, -7], [3, -8, -1, -1]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (3, [[1, -7, -7, -7], [3, -8, -1, -1]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

54. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=4, exact integer arithmetic function f(k)=a*k+b with a=4, b=-9. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(4, [[1, -5, -5, -5], [2, -6, -1, -1], [4, 1, 7, 7]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (4, [[1, -5, -5, -5], [2, -6, -1, -1], [4, 1, 7, 7]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

55. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=5, exact integer arithmetic function f(k)=a*k+b with a=5, b=-8. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(5, [[1, -3, -3, -3], [5, 14, 17, 17]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (5, [[1, -3, -3, -3], [5, 14, 17, 17]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

56. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=6, exact integer arithmetic function f(k)=a*k+b with a=1, b=-7. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(6, [[1, -6, -6, -6], [2, -11, -5, -5], [3, -10, -4, -4], [6, -16, -1, -1]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (6, [[1, -6, -6, -6], [2, -11, -5, -5], [3, -10, -4, -4], [6, -16, -1, -1]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

57. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=7, exact integer arithmetic function f(k)=a*k+b with a=2, b=-6. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(7, [[1, -4, -4, -4], [7, 4, 8, 8]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (7, [[1, -4, -4, -4], [7, 4, 8, 8]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

58. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=8, exact integer arithmetic function f(k)=a*k+b with a=3, b=-5. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(8, [[1, -2, -2, -2], [2, -1, 1, 1], [4, 6, 7, 7], [8, 25, 19, 19]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (8, [[1, -2, -2, -2], [2, -1, 1, 1], [4, 6, 7, 7], [8, 25, 19, 19]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

59. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=9, exact integer arithmetic function f(k)=a*k+b with a=4, b=-4. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(9, [[1, 0, 0, 0], [3, 8, 8, 8], [9, 40, 32, 32]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (9, [[1, 0, 0, 0], [3, 8, 8, 8], [9, 40, 32, 32]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

60. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=10, exact integer arithmetic function f(k)=a*k+b with a=5, b=-3. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(10, [[1, 2, 2, 2], [2, 9, 7, 7], [5, 24, 22, 22], [10, 78, 47, 47]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (10, [[1, 2, 2, 2], [2, 9, 7, 7], [5, 24, 22, 22], [10, 78, 47, 47]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

61. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=11, exact integer arithmetic function f(k)=a*k+b with a=1, b=-2. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(11, [[1, -1, -1, -1], [11, 8, 9, 9]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (11, [[1, -1, -1, -1], [11, 8, 9, 9]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

62. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=12, exact integer arithmetic function f(k)=a*k+b with a=2, b=-1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(12, [[1, 1, 1, 1], [2, 4, 3, 3], [3, 6, 5, 5], [4, 11, 7, 7], [6, 20, 11, 11], [12, 50, 23, 23]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (12, [[1, 1, 1, 1], [2, 4, 3, 3], [3, 6, 5, 5], [4, 11, 7, 7], [6, 20, 11, 11], [12, 50, 23, 23]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

63. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=13, exact integer arithmetic function f(k)=a*k+b with a=3, b=0. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(13, [[1, 3, 3, 3], [13, 42, 39, 39]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (13, [[1, 3, 3, 3], [13, 42, 39, 39]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

64. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=14, exact integer arithmetic function f(k)=a*k+b with a=4, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(14, [[1, 5, 5, 5], [2, 14, 9, 9], [7, 34, 29, 29], [14, 100, 57, 57]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (14, [[1, 5, 5, 5], [2, 14, 9, 9], [7, 34, 29, 29], [14, 100, 57, 57]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

65. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=15, exact integer arithmetic function f(k)=a*k+b with a=5, b=2. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(15, [[1, 7, 7, 7], [3, 24, 17, 17], [5, 34, 27, 27], [15, 128, 77, 77]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (15, [[1, 7, 7, 7], [3, 24, 17, 17], [5, 34, 27, 27], [15, 128, 77, 77]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

66. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=16, exact integer arithmetic function f(k)=a*k+b with a=1, b=3. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(16, [[1, 4, 4, 4], [2, 9, 5, 5], [4, 16, 7, 7], [8, 27, 11, 11], [16, 46, 19, 19]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (16, [[1, 4, 4, 4], [2, 9, 5, 5], [4, 16, 7, 7], [8, 27, 11, 11], [16, 46, 19, 19]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

67. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=17, exact integer arithmetic function f(k)=a*k+b with a=2, b=4. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(17, [[1, 6, 6, 6], [17, 44, 38, 38]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (17, [[1, 6, 6, 6], [17, 44, 38, 38]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

68. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=18, exact integer arithmetic function f(k)=a*k+b with a=3, b=5. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(18, [[1, 8, 8, 8], [2, 19, 11, 11], [3, 22, 14, 14], [6, 56, 23, 23], [9, 54, 32, 32], [18, 147, 59, 59]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (18, [[1, 8, 8, 8], [2, 19, 11, 11], [3, 22, 14, 14], [6, 56, 23, 23], [9, 54, 32, 32], [18, 147, 59, 59]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

69. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=19, exact integer arithmetic function f(k)=a*k+b with a=4, b=6. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(19, [[1, 10, 10, 10], [19, 92, 82, 82]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (19, [[1, 10, 10, 10], [19, 92, 82, 82]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

70. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=20, exact integer arithmetic function f(k)=a*k+b with a=5, b=7. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(20, [[1, 12, 12, 12], [2, 29, 17, 17], [4, 56, 27, 27], [5, 44, 32, 32], [10, 118, 57, 57], [20, 252, 107, 107]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (20, [[1, 12, 12, 12], [2, 29, 17, 17], [4, 56, 27, 27], [5, 44, 32, 32], [10, 118, 57, 57], [20, 252, 107, 107]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

71. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=21, exact integer arithmetic function f(k)=a*k+b with a=1, b=8. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(21, [[1, 9, 9, 9], [3, 20, 11, 11], [7, 24, 15, 15], [21, 64, 29, 29]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (21, [[1, 9, 9, 9], [3, 20, 11, 11], [7, 24, 15, 15], [21, 64, 29, 29]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

72. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=22, exact integer arithmetic function f(k)=a*k+b with a=2, b=9. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(22, [[1, 11, 11, 11], [2, 24, 13, 13], [11, 42, 31, 31], [22, 108, 53, 53]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (22, [[1, 11, 11, 11], [2, 24, 13, 13], [11, 42, 31, 31], [22, 108, 53, 53]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

73. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=23, exact integer arithmetic function f(k)=a*k+b with a=3, b=10. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(23, [[1, 13, 13, 13], [23, 92, 79, 79]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (23, [[1, 13, 13, 13], [23, 92, 79, 79]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

74. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=24, exact integer arithmetic function f(k)=a*k+b with a=4, b=11. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(24, [[1, 15, 15, 15], [2, 34, 19, 19], [3, 38, 23, 23], [4, 61, 27, 27], [6, 92, 35, 35], [8, 104, 43, 43], [12, 178, 59, 59], [24, 328, 107, 107]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (24, [[1, 15, 15, 15], [2, 34, 19, 19], [3, 38, 23, 23], [4, 61, 27, 27], [6, 92, 35, 35], [8, 104, 43, 43], [12, 178, 59, 59], [24, 328, 107, 107]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

75. Every Divisor Row Recovers Its Original Arithmetic Value

Use positive integer n=25, exact integer arithmetic function f(k)=a*k+b with a=5, b=12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).

View answer and explanation

(25, [[1, 17, 17, 17], [5, 54, 37, 37], [25, 191, 137, 137]], Yes, finite_divisor_poset_roundtrip)

For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (25, [[1, 17, 17, 17], [5, 54, 37, 37], [25, 191, 137, 137]], Yes, finite_divisor_poset_roundtrip).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-roundtrip-table · candidate

76. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=6, exact integer arithmetic function f(k)=a*k+b with a=1, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(6, [1, 2, 3, 6], 1, [2, 3, 6], 7, -9, 7, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (6, [1, 2, 3, 6], 1, [2, 3, 6], 7, -9, 7, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

77. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=12, exact integer arithmetic function f(k)=a*k+b with a=2, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(12, [1, 2, 3, 4, 6, 12], 1, [2, 3, 4, 6, 12], 25, -37, 25, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (12, [1, 2, 3, 4, 6, 12], 1, [2, 3, 4, 6, 12], 25, -37, 25, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

78. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=20, exact integer arithmetic function f(k)=a*k+b with a=3, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(20, [1, 2, 4, 5, 10, 20], 1, [2, 4, 5, 10, 20], 61, -71, 61, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (20, [1, 2, 4, 5, 10, 20], 1, [2, 4, 5, 10, 20], 61, -71, 61, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

79. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=30, exact integer arithmetic function f(k)=a*k+b with a=4, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(30, [1, 2, 3, 5, 6, 10, 15, 30], 1, [2, 3, 5, 6, 10, 15, 30], 121, -175, 121, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (30, [1, 2, 3, 5, 6, 10, 15, 30], 1, [2, 3, 5, 6, 10, 15, 30], 121, -175, 121, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

80. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=42, exact integer arithmetic function f(k)=a*k+b with a=5, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(42, [1, 2, 3, 6, 7, 14, 21, 42], 1, [2, 3, 6, 7, 14, 21, 42], 211, -277, 211, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (42, [1, 2, 3, 6, 7, 14, 21, 42], 1, [2, 3, 6, 7, 14, 21, 42], 211, -277, 211, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

81. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=56, exact integer arithmetic function f(k)=a*k+b with a=6, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(56, [1, 2, 4, 7, 8, 14, 28, 56], 1, [2, 4, 7, 8, 14, 28, 56], 337, -391, 337, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (56, [1, 2, 4, 7, 8, 14, 28, 56], 1, [2, 4, 7, 8, 14, 28, 56], 337, -391, 337, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

82. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=72, exact integer arithmetic function f(k)=a*k+b with a=7, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(72, [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72], 1, [2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72], 505, -872, 505, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (72, [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72], 1, [2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72], 505, -872, 505, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

83. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=90, exact integer arithmetic function f(k)=a*k+b with a=8, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(90, [1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90], 1, [2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90], 721, -1163, 721, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (90, [1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90], 1, [2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90], 721, -1163, 721, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

84. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=110, exact integer arithmetic function f(k)=a*k+b with a=9, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(110, [1, 2, 5, 10, 11, 22, 55, 110], 1, [2, 5, 10, 11, 22, 55, 110], 991, -961, 991, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (110, [1, 2, 5, 10, 11, 22, 55, 110], 1, [2, 5, 10, 11, 22, 55, 110], 991, -961, 991, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

85. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=132, exact integer arithmetic function f(k)=a*k+b with a=10, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(132, [1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132], 1, [2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132], 1321, -2051, 1321, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (132, [1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132], 1, [2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132], 1321, -2051, 1321, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

86. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=156, exact integer arithmetic function f(k)=a*k+b with a=11, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(156, [1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156], 1, [2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156], 1717, -2607, 1717, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (156, [1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156], 1, [2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156], 1717, -2607, 1717, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

87. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=182, exact integer arithmetic function f(k)=a*k+b with a=12, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(182, [1, 2, 7, 13, 14, 26, 91, 182], 1, [2, 7, 13, 14, 26, 91, 182], 2185, -1855, 2185, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (182, [1, 2, 7, 13, 14, 26, 91, 182], 1, [2, 7, 13, 14, 26, 91, 182], 2185, -1855, 2185, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

88. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=210, exact integer arithmetic function f(k)=a*k+b with a=13, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(210, [1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210], 1, [2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210], 2731, -4773, 2731, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (210, [1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210], 1, [2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210], 2731, -4773, 2731, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

89. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=240, exact integer arithmetic function f(k)=a*k+b with a=14, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(240, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240], 1, [2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240], 3361, -7075, 3361, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (240, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240], 1, [2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240], 3361, -7075, 3361, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

90. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=272, exact integer arithmetic function f(k)=a*k+b with a=15, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(272, [1, 2, 4, 8, 16, 17, 34, 68, 136, 272], 1, [2, 4, 8, 16, 17, 34, 68, 136, 272], 4081, -4299, 4081, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (272, [1, 2, 4, 8, 16, 17, 34, 68, 136, 272], 1, [2, 4, 8, 16, 17, 34, 68, 136, 272], 4081, -4299, 4081, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

91. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=306, exact integer arithmetic function f(k)=a*k+b with a=16, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(306, [1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306], 1, [2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306], 4897, -6347, 4897, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (306, [1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306], 1, [2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306], 4897, -6347, 4897, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

92. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=342, exact integer arithmetic function f(k)=a*k+b with a=17, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(342, [1, 2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342], 1, [2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342], 5815, -7457, 5815, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (342, [1, 2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342], 1, [2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342], 5815, -7457, 5815, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

93. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=380, exact integer arithmetic function f(k)=a*k+b with a=18, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(380, [1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380], 1, [2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380], 6841, -8291, 6841, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (380, [1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380], 1, [2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380], 6841, -8291, 6841, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

94. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=420, exact integer arithmetic function f(k)=a*k+b with a=19, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(420, [1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420], 1, [2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420], 7981, -17579, 7981, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (420, [1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420], 1, [2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420], 7981, -17579, 7981, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

95. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=462, exact integer arithmetic function f(k)=a*k+b with a=20, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(462, [1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 1, [2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 9241, -13815, 9241, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (462, [1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 1, [2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 9241, -13815, 9241, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

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Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

96. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=506, exact integer arithmetic function f(k)=a*k+b with a=21, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(506, [1, 2, 11, 22, 23, 46, 253, 506], 1, [2, 11, 22, 23, 46, 253, 506], 10627, -7525, 10627, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (506, [1, 2, 11, 22, 23, 46, 253, 506], 1, [2, 11, 22, 23, 46, 253, 506], 10627, -7525, 10627, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

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Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

97. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=552, exact integer arithmetic function f(k)=a*k+b with a=22, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(552, [1, 2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552], 1, [2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552], 12145, -19551, 12145, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (552, [1, 2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552], 1, [2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552], 12145, -19551, 12145, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

98. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=600, exact integer arithmetic function f(k)=a*k+b with a=23, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(600, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600], 1, [2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600], 13801, -29003, 13801, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (600, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600], 1, [2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600], 13801, -29003, 13801, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

99. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=650, exact integer arithmetic function f(k)=a*k+b with a=24, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(650, [1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650], 1, [2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650], 15601, -15659, 15601, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (650, [1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650], 1, [2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650], 15601, -15659, 15601, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

100. An Omitted Divisor Must Be a Discriminative Missing-Term Witness

Use positive integer n=702, exact integer arithmetic function f(k)=a*k+b with a=25, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).

View answer and explanation

(702, [1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702], 1, [2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702], 17551, -24465, 17551, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)

Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (702, [1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702], 1, [2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702], 17551, -24465, 17551, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required).

Reviewed step-by-step derivation

Mobius inversion on finite divisor sets · qf-mod41-mobinv-incomplete-divisor-boundary · candidate

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