slug: squarefree-mean-partition
bibkey: wiseman2023a360070
doi: null
url: https://oeis.org/A360070
triage: theorem
motivation_gids:
  - D5/S3/Combinatorics/SquarefreeMeanPartition.result

Squarefree Numbers Have No Partition Whose Parts and Multiplicities Agree in Mean

Problem

OEIS A360070, at revision #18 of Jan 29 2023, by Gus Wiseman, Jan 27 2023:

Numbers for which there exists an integer partition such that the parts have the same mean as the multiplicities.

Conjecture: No term > 1 is squarefree.

The entry’s worked example:

A partition of 20 with the same mean as its multiplicities is (5,4,3,2,1,1,1,1,1,1), so 20 is in the sequence.

Motivation

The frozen theorem D5/S3/Combinatorics/SquarefreeMeanPartition.result settles the comment.

Gap

Issue 9539 records the screen carried out before the work. A360070 appears nowhere under Problems/, D5/ or docs/ in this repository. It appears as one note-only row in the 2026-09-10 triage record in Library/Words/, where the same argument was sketched and the row was set aside for high bind risk. That disposition is reversed here: the risk named there is about the proof shape, which is reported as measured, and is not by itself a reason to leave a named external conjecture unjudged. A web search for a published proof returned only unrelated partition literature. Citation indices were not exhaustively reachable, so this is a bounded negative finding.

Route

Present a partition by its set D of distinct parts together with a multiplicity function m. Let M be the number of parts, the sum of the multiplicities, and k = |D| the number of distinct parts. The mean of the parts is n / M, the mean of the multiplicities is M / k, so the source’s condition is n * k = M ^ 2 after clearing denominators. Both denominators are positive for a partition of a positive number, so nothing is lost and the statement stays inside the naturals. The worked example is this equation at M = 10, k = 5: 20 * 5 = 100 = 10 ^ 2, both means being two.

Suppose n > 1 is squarefree and such a partition exists.

One. n * k = M ^ 2 gives n ∣ M ^ 2; with n squarefree this gives n ∣ M, say M = n * j.

Two. Substituting and cancelling the positive n leaves k = n * j ^ 2.

Three. Every multiplicity is at least one, so the number of distinct parts is at most the number of parts, k ≤ M, that is n * j ^ 2 ≤ n * j, forcing j ≤ 1; and M ≥ 1 since D is nonempty, so j = 1 and M = k = n.

Four. k = M with every multiplicity at least one forces every multiplicity to be exactly one. The total is then the sum of the distinct parts, n of them, each at least one. If one of them were at least two the total would exceed n. So every part is one, so D ⊆ {1} and k ≤ 1, contradicting k = n > 1.

Squarefreeness enters at exactly one place, step one, and that is why the conclusion is about squarefreeness rather than about n being composite.

Falsifier

The literal sentence fails at n = 1: the partition (1) has one part and one distinct part, both means equal to one, and 1 is squarefree; the entry’s data has a(1) = 1. This is why the comment says term > 1 and why the statement carries that bound. The sister entry A360068 carries the same sentence without the bound and is false as literally written for this reason — a boundary observation from the definition and the data, not a report of a published refutation.

Reading the mean of the multiplicities as the mean over parts rather than over distinct parts would give a different equation and a different sequence; the worked example pins the reading.

Evidence

Enumerating every partition and testing n * k = M ^ 2 reproduces the entry’s data line term by term up to n = 60: 1, 4, 8, 9, 12, 16, 18, 20, 25, 27, 32, 36, 45, 48, 49, 50, 54. The worked example checks out as recorded above.

Independently of any partition, the arithmetic alone was swept: for every squarefree n < 2000, the pairs (M, k) with n * k = M ^ 2 and k ≤ M were listed, and the only solution is the diagonal M = k = n — exactly the case step four kills. No squarefree n > 1 appears among the computed terms.

Triage

theorem; Tier 1 named external open question, preregistered in issue 9539 before the work. 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 every n and every partition.

ASSUMED-UNVERIFIED

The literature screen is bounded. The entry was read in full at revision #18. Citation indices and printed sources were not exhaustively reachable, so no worldwide priority claim is made. The weight is stated plainly: the argument is elementary, and what is settled is that the sentence sat on the entry unjudged since January 2023.