Complete Prime Multiplicities, Positive-Domain Mobius Values and Divisor Sums
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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.
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Mobius function · qf-mod40-mobius-domain-boundary · candidate
Canonical question JSONL · English JSONL · Validation report