slug: coprime-divisor-tuples bibkey: wiseman2021a062319 doi: null url: https://oeis.org/A062319 triage: theorem motivation_gids:

• D5/S3/Arith/CoprimeDivisorTuples.result

Pairwise Coprime Ordered Tuples of Divisors

Problem

OEIS A062319, “Number of divisors of n^n, or of A000312(n)”, carries a comment by Gus Wiseman dated May 02 2021 and still standing at revision #60 of Aug 30 2026:

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.

The comment goes on to print the tuples counted for n = 1 through n = 5. Those lists fix two readings the words leave open: the value one may occur and may repeat, so the constant tuple of ones counts, and the order of the coordinates matters, so (1,2) and (2,1) count separately.

Motivation

The frozen theorem D5/S3/Arith/CoprimeDivisorTuples.result settles the comment in the affirmative for every n ≥ 1, and does so by way of a statement in which the tuple length is a free parameter, so that the comment is the diagonal case.

Gap

Issue 9455 records the screen carried out before the work. A062319 appears nowhere under Problems/, D5/, Library/ or docs/ in this repository except in the triage record Library/Words/oeis2026triage0910.md, where it was parked as note-only, not dropped, on the ground that the count sits close to existing prime factorisation and divisor cardinality theorems and that the wider literature on coprime tuples had not been checked. That check is what this dossier closes.

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. Its two nearest cross-references carry no proof: the unordered analogue A343654, “pairwise coprime n-multisets of divisors of n”, lists only further cross-references, and the array A343656, of which this sequence is the diagonal, carries only the divisor count formula. 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.

Route

Both sides equal ∏ p ∈ primeFactors n, (1 + n · m_p), where m_p is the exponent of p in n.

The right side is immediate. The exponent of p in n^n is n · m_p, so the divisor count of n^n is that product. The entry already records the same product as a formula for its own terms, so the settlement rests on the tuple side.

The tuple side factors over the primes of n. Pairwise coprimality says exactly that for each prime p dividing n, at most one coordinate of the tuple is divisible by p. So the exponent matrix of a tuple splits into one column per prime, and each column is a vector of bounded entries that is zero except in at most one place. A column is therefore either identically zero, one possibility, or determined by a coordinate together with an exponent between 1 and m_p, which is n · m_p possibilities. Multiplying over the primes gives the product, with no appeal to multiplicativity as an external theorem.

The formal argument builds three bijections and composes them. A divisor of n is identified with its bounded exponent vector; a pair of divisors is coprime exactly when no prime has positive exponent in both; and a sparse column is identified with an option type over the pairs consisting of a coordinate and a positive exponent.

The tuple length must be a parameter independent of n. The one-parameter form, in which the length is tied to the modulus, is not multiplicative in n, so the induction that proves the product formula has to be carried out at fixed length.

Falsifier

A tuple that is pairwise coprime yet has two coordinates divisible by a common prime of n, or a prime column with a second nonzero entry, would break the splitting. A different reading of the entry, one that forbids repeated ones or identifies a tuple with its reordering, would give a different left-hand count; the worked lists on the entry exclude both readings, since (1,1,1,1,1) is listed and (1,2) and (2,1) are listed separately.

Evidence

Brute-force enumeration of the tuples for n = 1..7 gives 1, 3, 4, 9, 6, 49, 8, agreeing term by term with the divisor counts of n^n, with the product ∏ (1 + n · m_p), and with the entry’s own data line. The product formula agrees with the divisor count of n^n also at n = 8, 9, 10, 12, 16, 18, 24, 30, 36, 60. The tuple side is exponential to enumerate, so it was checked directly only to n = 7; the proof covers the rest.

The mathematics is elementary. What the settlement records is that a named published assertion had not been judged, not that the argument is deep.

Triage

theorem; Tier 1 named external open question, preregistered in issue 9455 before the work. The admission basis is open-problem-resolution. The computational use is none: no declaration is a bounded enumeration, a checker, a numeric reduction or a certified instance, and the delivered statement is universally quantified over all positive n.

ASSUMED-UNVERIFIED

The literature screen is bounded. The entry was read in full at revision #60; A343654, A343656, the arXiv corpus of 2608.11941, the epoch-research/LeanOpenProblems-results runs and the google-deepmind/alphaproof-nexus-results Lean files were checked; arXiv abstract searches for pairwise coprime tuples of divisors were run. Citation indices and printed sources were not exhaustively reachable, so no worldwide priority claim is made.