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Metadata 
bibkey: ianakiev2018a319927
authors: Ivan N. Ianakiev
year: 2018
title: "OEIS A319927, 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"
doi: null
url: https://oeis.org/A319927
claim: "A319927 %N: 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. A319927 %C: 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). A319927 %C: 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 A319927 %A: _Ivan N. Ianakiev_, Oct 02 2018"
strata_touched:
  - D5/S0/Certificates/IanakievOddNonunitaryPowerSumRefutation
license: citation-only
triage: anchor

OEIS A319927

The NAME of A319927 states:

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.

The COMMENT conjecture is printed as:

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 states:

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 AUTHOR line is:

Ivan N. Ianakiev, Oct 02 2018

The PARI program defines isok(n) using if (suo, ...), so sequence membership includes a nonzero odd non-unitary square-sum guard. The conjecture fails at the sequence member 9216 for p = 1, since the odd sum is 3 and 3 does not divide 16361; Munn’s 2020 edit discussion asked whether 9216 was a counterexample at p = 3, but the entry never answered that question. This delivery makes no publication-priority claim.

Verified locator

  • URL: https://oeis.org/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): 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).
  • PARI membership guard (verbatim): isok(n) = my(suo = suo2(n)); if (suo, (su2(n) % suo) == 0);
  • COMMENT by Peter Munn (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
  • AUTHOR (verbatim): Ivan N. Ianakiev, Oct 02 2018

The entry and its edit history were read on 2026-09-14. Revisions #1 through #30 leave the conjecture unchanged; revision #30, dated 2025-02-16, changes only a MathWorld link from HTTP to HTTPS. Revision #21’s edit discussion contains Munn’s unanswered question about 9216 at p = 3.

Scope of the formal work

The result refutes the displayed universal conjecture only. It does not claim the 2022 remark that p*a(n) is a term, any statement about A034444 or A048105, exhaustive literature coverage, or publication priority.

ASSUMED-UNVERIFIED

OpenAlex returned HTTP 429 and Google Scholar returned a CAPTCHA. Those surfaces were not verified, and the bounded literature search does not establish exhaustive coverage.