# Complete Prime Multiplicities, Positive-Domain Mobius Values and Divisor Sums

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261109-r06-mobius-function-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=2. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(2, [2], [2], -1, squarefree_parity_sign)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (2, [2], [2], -1, squarefree_parity_sign).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0001/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 2. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=6. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(6, [2, 3], [2, 3], 1, squarefree_parity_sign)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (6, [2, 3], [2, 3], 1, squarefree_parity_sign).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0002/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 3. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=12. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(12, [2, 2, 3], [2, 3], 0, square_prime_factor_gives_zero)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (12, [2, 2, 3], [2, 3], 0, square_prime_factor_gives_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0003/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 4. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=20. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(20, [2, 2, 5], [2, 5], 0, square_prime_factor_gives_zero)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (20, [2, 2, 5], [2, 5], 0, square_prime_factor_gives_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0004/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 5. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=30. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(30, [2, 3, 5], [2, 3, 5], -1, squarefree_parity_sign)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (30, [2, 3, 5], [2, 3, 5], -1, squarefree_parity_sign).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0005/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 6. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=42. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(42, [2, 3, 7], [2, 3, 7], -1, squarefree_parity_sign)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (42, [2, 3, 7], [2, 3, 7], -1, squarefree_parity_sign).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0006/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 7. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=56. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(56, [2, 2, 2, 7], [2, 7], 0, square_prime_factor_gives_zero)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (56, [2, 2, 2, 7], [2, 7], 0, square_prime_factor_gives_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0007/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 8. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=72. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(72, [2, 2, 2, 3, 3], [2, 3], 0, square_prime_factor_gives_zero)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (72, [2, 2, 2, 3, 3], [2, 3], 0, square_prime_factor_gives_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0008/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 9. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=90. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(90, [2, 3, 3, 5], [2, 3, 5], 0, square_prime_factor_gives_zero)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (90, [2, 3, 3, 5], [2, 3, 5], 0, square_prime_factor_gives_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0009/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 10. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=110. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(110, [2, 5, 11], [2, 5, 11], -1, squarefree_parity_sign)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (110, [2, 5, 11], [2, 5, 11], -1, squarefree_parity_sign).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0010/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 11. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=132. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(132, [2, 2, 3, 11], [2, 3, 11], 0, square_prime_factor_gives_zero)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (132, [2, 2, 3, 11], [2, 3, 11], 0, square_prime_factor_gives_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0011/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 12. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=156. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(156, [2, 2, 3, 13], [2, 3, 13], 0, square_prime_factor_gives_zero)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (156, [2, 2, 3, 13], [2, 3, 13], 0, square_prime_factor_gives_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0012/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 13. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=182. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(182, [2, 7, 13], [2, 7, 13], -1, squarefree_parity_sign)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (182, [2, 7, 13], [2, 7, 13], -1, squarefree_parity_sign).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0013/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 14. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=210. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(210, [2, 3, 5, 7], [2, 3, 5, 7], 1, squarefree_parity_sign)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (210, [2, 3, 5, 7], [2, 3, 5, 7], 1, squarefree_parity_sign).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0014/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 15. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=240. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(240, [2, 2, 2, 2, 3, 5], [2, 3, 5], 0, square_prime_factor_gives_zero)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (240, [2, 2, 2, 2, 3, 5], [2, 3, 5], 0, square_prime_factor_gives_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0015/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 16. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=272. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(272, [2, 2, 2, 2, 17], [2, 17], 0, square_prime_factor_gives_zero)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (272, [2, 2, 2, 2, 17], [2, 17], 0, square_prime_factor_gives_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0016/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 17. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=306. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(306, [2, 3, 3, 17], [2, 3, 17], 0, square_prime_factor_gives_zero)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (306, [2, 3, 3, 17], [2, 3, 17], 0, square_prime_factor_gives_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0017/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 18. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=342. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(342, [2, 3, 3, 19], [2, 3, 19], 0, square_prime_factor_gives_zero)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (342, [2, 3, 3, 19], [2, 3, 19], 0, square_prime_factor_gives_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0018/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 19. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=380. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(380, [2, 2, 5, 19], [2, 5, 19], 0, square_prime_factor_gives_zero)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (380, [2, 2, 5, 19], [2, 5, 19], 0, square_prime_factor_gives_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0019/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 20. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=420. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(420, [2, 2, 3, 5, 7], [2, 3, 5, 7], 0, square_prime_factor_gives_zero)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (420, [2, 2, 3, 5, 7], [2, 3, 5, 7], 0, square_prime_factor_gives_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0020/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 21. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=462. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(462, [2, 3, 7, 11], [2, 3, 7, 11], 1, squarefree_parity_sign)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (462, [2, 3, 7, 11], [2, 3, 7, 11], 1, squarefree_parity_sign).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0021/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 22. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=506. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(506, [2, 11, 23], [2, 11, 23], -1, squarefree_parity_sign)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (506, [2, 11, 23], [2, 11, 23], -1, squarefree_parity_sign).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0022/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 23. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=552. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(552, [2, 2, 2, 3, 23], [2, 3, 23], 0, square_prime_factor_gives_zero)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (552, [2, 2, 2, 3, 23], [2, 3, 23], 0, square_prime_factor_gives_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0023/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 24. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=600. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(600, [2, 2, 2, 3, 5, 5], [2, 3, 5], 0, square_prime_factor_gives_zero)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (600, [2, 2, 2, 3, 5, 5], [2, 3, 5], 0, square_prime_factor_gives_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0024/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 25. Complete Prime Multiplicities Determine the Mobius Value

Use positive integer n=650. Give (n, prime_multiset, distinct_primes, mu, certificate).

View answer and explanation(650, [2, 5, 5, 13], [2, 5, 13], 0, square_prime_factor_gives_zero)

Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (650, [2, 5, 5, 13], [2, 5, 13], 0, square_prime_factor_gives_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0025/)

Mobius function · qf-mod40-mobius-canonical-value · candidate





## 26. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=4, supplied factors=[2, 2] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(4, [2, 2], Yes, 0, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (4, [2, 2], Yes, 0, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0026/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 27. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=6, supplied factors=[2, 3] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(6, [2, 3], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (6, [2, 3], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0027/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 28. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=10, supplied factors=[2, 5] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(10, [2, 5], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (10, [2, 5], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0028/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 29. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=14, supplied factors=[2, 7] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(14, [2, 7], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (14, [2, 7], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0029/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 30. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=22, supplied factors=[2, 11] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(22, [2, 11], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (22, [2, 11], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0030/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 31. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=26, supplied factors=[2, 13] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(26, [2, 13], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (26, [2, 13], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0031/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 32. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=34, supplied factors=[2, 17] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(34, [2, 17], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (34, [2, 17], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0032/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 33. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=38, supplied factors=[2, 19] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(38, [2, 19], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (38, [2, 19], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0033/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 34. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=46, supplied factors=[2, 23] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(46, [2, 23], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (46, [2, 23], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0034/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 35. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=58, supplied factors=[2, 29] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(58, [2, 29], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (58, [2, 29], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0035/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 36. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=62, supplied factors=[2, 31] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(62, [2, 31], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (62, [2, 31], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0036/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 37. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=74, supplied factors=[2, 37] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(74, [2, 37], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (74, [2, 37], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0037/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 38. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=82, supplied factors=[2, 41] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(82, [2, 41], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (82, [2, 41], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0038/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 39. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=86, supplied factors=[2, 43] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(86, [2, 43], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (86, [2, 43], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0039/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 40. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=94, supplied factors=[2, 47] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(94, [2, 47], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (94, [2, 47], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0040/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 41. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=106, supplied factors=[2, 53] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(106, [2, 53], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (106, [2, 53], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0041/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 42. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=118, supplied factors=[2, 59] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(118, [2, 59], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (118, [2, 59], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0042/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 43. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=122, supplied factors=[2, 61] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(122, [2, 61], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (122, [2, 61], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0043/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 44. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=134, supplied factors=[2, 67] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(134, [2, 67], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (134, [2, 67], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0044/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 45. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=142, supplied factors=[2, 71] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(142, [2, 71], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (142, [2, 71], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0045/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 46. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=146, supplied factors=[2, 73] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(146, [2, 73], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (146, [2, 73], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0046/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 47. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=158, supplied factors=[2, 79] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(158, [2, 79], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (158, [2, 79], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0047/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 48. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=166, supplied factors=[2, 83] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(166, [2, 83], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (166, [2, 83], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0048/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 49. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=178, supplied factors=[2, 89] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(178, [2, 89], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (178, [2, 89], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0049/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 50. A Matching Composite Product Does Not Qualify as a Prime Factorization

Use positive integer n=194, supplied factors=[2, 97] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).

View answer and explanation(194, [2, 97], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)

Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (194, [2, 97], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0050/)

Mobius function · qf-mod40-mobius-factorization-audit · candidate





## 51. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=1. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(1, [[1, 1]], 1, 1, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (1, [[1, 1]], 1, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0051/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 52. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=2. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(2, [[1, 1], [2, -1]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (2, [[1, 1], [2, -1]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0052/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 53. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=3. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(3, [[1, 1], [3, -1]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (3, [[1, 1], [3, -1]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0053/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 54. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=4. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(4, [[1, 1], [2, -1], [4, 0]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (4, [[1, 1], [2, -1], [4, 0]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0054/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 55. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=5. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(5, [[1, 1], [5, -1]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (5, [[1, 1], [5, -1]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0055/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 56. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=6. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(6, [[1, 1], [2, -1], [3, -1], [6, 1]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (6, [[1, 1], [2, -1], [3, -1], [6, 1]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0056/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 57. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=7. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(7, [[1, 1], [7, -1]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (7, [[1, 1], [7, -1]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0057/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 58. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=8. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(8, [[1, 1], [2, -1], [4, 0], [8, 0]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (8, [[1, 1], [2, -1], [4, 0], [8, 0]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0058/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 59. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=9. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(9, [[1, 1], [3, -1], [9, 0]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (9, [[1, 1], [3, -1], [9, 0]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0059/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 60. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=10. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(10, [[1, 1], [2, -1], [5, -1], [10, 1]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (10, [[1, 1], [2, -1], [5, -1], [10, 1]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0060/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 61. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=11. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(11, [[1, 1], [11, -1]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (11, [[1, 1], [11, -1]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0061/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 62. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=12. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(12, [[1, 1], [2, -1], [3, -1], [4, 0], [6, 1], [12, 0]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (12, [[1, 1], [2, -1], [3, -1], [4, 0], [6, 1], [12, 0]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0062/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 63. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=13. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(13, [[1, 1], [13, -1]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (13, [[1, 1], [13, -1]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0063/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 64. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=14. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(14, [[1, 1], [2, -1], [7, -1], [14, 1]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (14, [[1, 1], [2, -1], [7, -1], [14, 1]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0064/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 65. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=15. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(15, [[1, 1], [3, -1], [5, -1], [15, 1]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (15, [[1, 1], [3, -1], [5, -1], [15, 1]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0065/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 66. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=16. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(16, [[1, 1], [2, -1], [4, 0], [8, 0], [16, 0]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (16, [[1, 1], [2, -1], [4, 0], [8, 0], [16, 0]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0066/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 67. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=17. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(17, [[1, 1], [17, -1]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (17, [[1, 1], [17, -1]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0067/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 68. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=18. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(18, [[1, 1], [2, -1], [3, -1], [6, 1], [9, 0], [18, 0]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (18, [[1, 1], [2, -1], [3, -1], [6, 1], [9, 0], [18, 0]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0068/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 69. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=19. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(19, [[1, 1], [19, -1]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (19, [[1, 1], [19, -1]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0069/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 70. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=20. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(20, [[1, 1], [2, -1], [4, 0], [5, -1], [10, 1], [20, 0]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (20, [[1, 1], [2, -1], [4, 0], [5, -1], [10, 1], [20, 0]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0070/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 71. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=21. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(21, [[1, 1], [3, -1], [7, -1], [21, 1]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (21, [[1, 1], [3, -1], [7, -1], [21, 1]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0071/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 72. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=22. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(22, [[1, 1], [2, -1], [11, -1], [22, 1]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (22, [[1, 1], [2, -1], [11, -1], [22, 1]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0072/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 73. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=23. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(23, [[1, 1], [23, -1]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (23, [[1, 1], [23, -1]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0073/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 74. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=24. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(24, [[1, 1], [2, -1], [3, -1], [4, 0], [6, 1], [8, 0], [12, 0], [24, 0]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (24, [[1, 1], [2, -1], [3, -1], [4, 0], [6, 1], [8, 0], [12, 0], [24, 0]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0074/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 75. All Positive Divisors Supply the Exact Mobius Sum Identity

Use positive integer n=25. Give (n, divisor_mu_rows, sum, target, equal).

View answer and explanation(25, [[1, 1], [5, -1], [25, 0]], 0, 0, Yes)

Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (25, [[1, 1], [5, -1], [25, 0]], 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0075/)

Mobius function · qf-mod40-mobius-divisor-sum · candidate





## 76. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=0. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(0, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (0, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0076/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 77. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-1. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-1, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-1, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0077/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 78. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-2. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-2, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-2, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0078/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 79. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-3. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-3, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-3, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0079/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 80. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-4. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-4, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-4, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0080/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 81. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-5. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-5, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-5, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0081/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 82. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-6. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-6, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-6, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0082/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 83. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-7. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-7, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-7, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0083/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 84. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-8. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-8, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-8, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0084/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 85. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-9. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-9, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-9, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0085/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 86. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-10. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-10, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-10, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0086/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 87. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-11. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-11, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-11, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0087/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 88. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-12. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-12, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-12, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0088/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 89. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-13. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-13, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-13, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0089/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 90. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-14. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-14, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-14, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0090/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 91. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-15. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-15, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-15, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0091/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 92. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-16. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-16, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-16, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0092/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 93. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-17. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-17, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-17, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0093/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 94. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-18. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-18, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-18, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0094/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 95. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-19. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-19, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-19, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0095/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 96. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-20. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-20, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-20, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0096/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 97. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-21. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-21, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-21, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0097/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 98. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-22. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-22, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-22, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0098/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 99. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-23. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-23, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-23, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0099/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate





## 100. Zero and Negative Integers Are Outside This Mobius Domain

Use nonpositive bounded integer n=-24. Give (n, in_positive_domain, mu_or_null, boundary).

View answer and explanation(-24, No, null, Mobius_domain_positive_integers_only)

Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-24, No, null, Mobius_domain_positive_integers_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r06-mobius-function-0100/)

Mobius function · qf-mod40-mobius-domain-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261109-r06-mobius-function-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261109-r06-mobius-function-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261109-r06-mobius-function-foundations/validation-report.json)