slug: oeis-a319927-odd-nonunitary-power-sum-refutation bibkey: ianakiev2018a319927 doi: null url: https://oeis.org/A319927 triage: theorem motivation_gids:

• D5/S0/Certificates/IanakievOddNonunitaryPowerSumRefutation

Refutation of the A319927 odd non-unitary power-sum conjecture

Problem

OEIS A319927, NAME (verbatim):

Numbers k such that the sum of the squares of the odd non-unitary divisors of k divides the sum of the squares of the non-unitary divisors of k.

COMMENT conjecture line (verbatim; Ivan N. Ianakiev, Oct 02 2018):

Conjecture: For any nonnegative integer power p the sum of the p-th powers of the odd non-unitary divisors of a(n) divides the sum of the p-th powers of the non-unitary divisors of a(n).

Peter Munn’s COMMENT (verbatim):

The start of this sequence, including the 58 terms currently shown in the data section, is consistent with a definition “nonsquarefree numbers not divisible by 4”, but some larger terms are divisible by 4: for example, a(1305) = 9216 = 2^10 * 3^2. - Peter Munn, Sep 21 2020

The PARI membership test (verbatim) is:

isok(n) = my(suo = suo2(n)); if (suo, (su2(n) % suo) == 0);

The AUTHOR line is:

Ivan N. Ianakiev, Oct 02 2018

The first terms printed in the DATA section are 9, 18, 25, 27, 45, 49, 50, 54, 63, 75, ....

For natural numbers, define nonunitaryDivisors(n) = Nat.divisors(n).filter(d => 1 < gcd(d,n/d)), where n/d is natural-number division and is exact for every divisor in Nat.divisors(n). Define S(k,n) as the sum of d^k over those divisors, and O(k,n) as the corresponding sum after the further filter d % 2 = 1. The source program’s nonzero guard is retained in member(n) := 0 < n and 0 < O(2,n) and O(2,n) divides S(2,n). The literal refuted statement is ∀ n : ℕ, member n → ∀ k : ℕ, O k n ∣ S k n.

The result does not assert the 2022 remark that p*a(n) is a term, and it does not assert anything about A034444 or A048105.

Motivation

Ianakiev’s 2018 comment states a universal divisibility conjecture for every nonnegative power at every sequence member. One explicit sequence member and power at which divisibility fails resolves that literal statement.

Gap

Preregistration issue #7646 and its probe report record searches dated September 14, 2026. OEIS revisions #1 through #30 leave the conjecture unchanged; revision #30, dated 2025-02-16, changes only a MathWorld link from HTTP to HTTPS. The discussion for revision #21 contains Peter Munn’s 2020-09-22 question, “Is a(1305) also a counterexample to the conjecture? … a(1305) with p = 3, to be more specific.” The entry never answered that question.

Exact searches returned 0 results on arXiv, 0 on MathOverflow, and 0 on Crossref. OpenAlex autocomplete returned 0 and /works returned HTTP 429 (ASSUMED-UNVERIFIED). GitHub returned two OEIS program mirrors and issue #7646 only. Google Scholar returned a CAPTCHA (ASSUMED-UNVERIFIED). No publication-priority or exhaustive-literature claim follows from these bounded surfaces.

Route

Finite evaluation gives O(2,9216)=9 and S(2,9216)=41243877, so 9216 satisfies the guarded sequence-membership predicate. At k=1, the same evaluation gives O(1,9216)=3 and S(1,9216)=16361; the latter leaves remainder 2 modulo 3. Instantiating the universal claim at n=9216 and k=1 therefore yields a contradiction.

Falsifier

A derivation of the literal universal claim would falsify this refutation. The result supplies member(9216) and the incompatible fact that O(1,9216) does not divide S(1,9216).

Evidence

• Lean module: D5/S0/Certificates/IanakievOddNonunitaryPowerSumRefutation.lean.
• Main theorem: result : Not claim, with exactly the std3 axioms propext, Classical.choice, and Quot.sound.
• The profiled Lean process took 4.86 seconds wall time and 641 milliseconds of cumulative type checking, with maximum resident set size 1,781,022,720 bytes. The two decide certificates accounted for the observed 674 milliseconds of tactic execution.
• The result uses finite evaluation to establish both member(9216) and the failed divisibility at k=1; no private helper declaration is present.

The orchestrator computed 9216 = 2^10*3^2, 33 divisors, 29 non-unitary divisors, unitary divisors {1,9,1024,9216}, and the single odd non-unitary divisor {3}. Its power-sum readings are S(0)=29, O(0)=1; S(1)=16361=3*5453+2, O(1)=3; S(2)=41243877=9*4582653, O(2)=9; S(3)=145108537091, O(3)=27, with remainder 17; and remainder 63 at k=4.

For n <= 10^5, the orchestrator found 14,211 members and five failures at k=1: 9216, 27648, 46080, 51200, 64512; every listed value also failed at k=3 and k=4. The probe independently reproduced these n <= 10^5 readings. The search seat reported a C++ scan through n <= 10^6 with 142,128 members, 54 failures at k=1, and least failure 9216.

These bounded computations support the single instance used by result. They do not assert a classification of all failures. Munn’s 2020 question about p=3 and the 2022 remark are not claimed, and neither are statements about A034444 or A048105.

Triage

theorem. The sequence member 9216 at k=1 refutes the literal universal conjecture. No publication-priority claim is made.

ASSUMED-UNVERIFIED

OpenAlex /works, Google Scholar, the search seat’s scan through 10^6, and all bounded searches beyond the single finite instance used by result are ASSUMED-UNVERIFIED. The literature search is bounded and does not establish exhaustive coverage or publication priority.