bibkey: wiseman2023a360070 authors: Gus Wiseman year: 2023 title: “OEIS A360070, Numbers for which there exists an integer partition such that the parts have the same mean as the multiplicities” doi: null url: https://oeis.org/A360070 claim: “Conjecture: No term > 1 is squarefree.” strata_touched:

• D5/S3/Combinatorics/SquarefreeMeanPartition license: citation-only triage: anchor

OEIS A360070

The entry lists the numbers admitting a partition whose parts and whose multiplicities have the same mean. Its terms begin 1, 4, 8, 9, 12, 16, 18, 20, 25, 27, 32, 36, 45, 48, 49, 50, 54, ….

Verified locator

• URL: https://oeis.org/A360070
• Locator: NAME, “Numbers for which there exists an integer partition such that the parts have the same mean as the multiplicities”; COMMENTS, “Conjecture: No term > 1 is squarefree.”
• Revision read: #18, Jan 29 2023. The comment stands and carries no answer.

Reading of the statement

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 and the mean of the multiplicities is M / k, so the condition is

n * k = M ^ 2

once denominators are cleared. Both denominators are positive for a partition of a positive number, so nothing is lost, and the condition stays inside the naturals.

The entry’s worked example fixes the reading: “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.” There M = 10 and k = 5, both means are two, and 20 * 5 = 100 = 10 ^ 2.

The boundary, which is real

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. The sister entry A360068 carries the same sentence without that bound and is false as literally written for this reason; that is a boundary observation from the definition and the data, not a report of a published refutation.

Scope of the recorded answer

The comment holds, and squarefreeness does all the work in four steps.

Since n * k = M ^ 2, n divides M ^ 2; with n squarefree this gives n ∣ M, say M = n * j. Substituting and cancelling the positive n leaves k = n * j ^ 2. 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, which forces j ≤ 1; and M ≥ 1, so j = 1 and M = k = n.

Then k = M with every multiplicity at least one forces every multiplicity to be exactly one, so the total is 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.

The prime powers show why the conclusion is about squarefreeness rather than about n being composite: the step that fails for a non-squarefree n is exactly n ∣ M ^ 2 ⟹ n ∣ M.

Bounded prior-resolution evidence

The entry was read in full at revision #18 and records no proof and no reference to one. 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 and no worldwide priority claim is made.