bibkey: wiseman2021a062319 authors: Gus Wiseman year: 2021 title: “OEIS A062319, Number of divisors of n^n, or of A000312(n)” doi: null url: https://oeis.org/A062319 claim: “Conjecture: The number of divisors of n^n equals the number of pairwise coprime ordered n-tuples of divisors of n. Confirmed up to n = 30. - Gus Wiseman, May 02 2021” strata_touched:

• D5/S3/Arith/CoprimeDivisorTuples license: citation-only triage: anchor

OEIS A062319

The entry lists the number of divisors of n^n. Its terms begin 1, 1, 3, 4, 9, 6, 49, 8, 25, 19, 121, 12, 325, … from n = 0.

Verified locator

• URL: https://oeis.org/A062319
• Locator: COMMENTS, “Conjecture: The number of divisors of n^n equals the number of pairwise coprime ordered n-tuples of divisors of n. Confirmed up to n = 30.”
• Revision read: #60, Aug 30 2026. The comment stands and carries no answer.

Reading of the statement

An ordered n-tuple of divisors of n is a function from the first n indices to the divisors of n; it is pairwise coprime when any two coordinates at distinct indices have greatest common divisor one. The entry’s own worked lists for n = 1 through n = 5 fix two readings that the words leave open: an entry equal to one may repeat, so (1,1,1,1,1) counts, and the order of the coordinates matters, so (1,2) and (2,1) count separately.

Scope of the recorded answer

The comment holds for every n ≥ 1. Both sides are the same product over the primes dividing n.

The right side is the shorter half. The exponent of a prime p in n^n is n times its exponent in n, so the divisor count of n^n is the product of n · m_p + 1 over the primes p of n, where m_p is the exponent of p in n. That product is already recorded on the entry as a formula for the sequence itself.

The left side is where the content sits. Pairwise coprimality of a tuple says exactly that for each prime p dividing n, at most one coordinate is divisible by p. So the tuple is determined by an independent choice at each prime: either no coordinate carries p, which is one possibility, or one of the n coordinates carries p to an exponent between 1 and m_p, which is n · m_p possibilities. The exponent matrix of the tuple therefore splits into one column per prime, each column being a vector that is zero except in at most one place, and the count is the same product.

The formal route makes that splitting explicit rather than appealing to multiplicativity. A divisor of n is identified with its bounded exponent vector; coprimality of two divisors becomes the statement that no prime has a positive exponent in both; and a sparse column is identified with either the empty choice or a pair consisting of a coordinate and a positive exponent. The tuple length is carried as a parameter independent of the modulus, because the count is a product over the primes of n only after the length is held fixed; the single-parameter form, with the length tied to n, is not multiplicative in n.

The statement generalises without extra work: for every length k and every n ≥ 1, the pairwise coprime ordered k-tuples of divisors of n number the product of k · m_p + 1 over the primes of n. The entry’s conjecture is the diagonal k = n.

Bounded prior-resolution evidence

The entry was read in full at revision #60 and records no proof and no reference to one. The unordered companion A343654, “pairwise coprime n-multisets of divisors of n”, carries only cross-references; the array A343656, of which this sequence is the diagonal, carries only the divisor count. The entry is absent from the 492-statement corpus of arXiv:2608.11941, from every run of the epoch-research/LeanOpenProblems-results collection, and from the Lean files of google-deepmind/alphaproof-nexus-results. Searches of arXiv abstracts for pairwise coprime tuples of divisors returned nothing. Citation indices and printed sources were not exhaustively reachable, so this is a bounded negative finding and no worldwide priority claim is made.