slug: oeis-a245786-integrality-gcd-record-refutation bibkey: krizek2014a245786 doi: null url: https://oeis.org/A245786 triage: theorem motivation_gids:

• D5/S0/Certificates/KrizekIntegralityGcdRecordRefutation

Refutation of the A245786 integrality-implies-gcd-record conjecture

Problem

OEIS A245786, NAME (verbatim):

Numbers n such that k(n) = (n/tau(n) + sigma(n)/n) is an integer.

COMMENT conjecture line (verbatim; Jaroslav Krizek, Aug 15 2014):

Conjecture: Subsequence of A216793.

OEIS A216793, NAME (verbatim; Michel Marcus, Sep 16 2012):

Numbers n such that gcd(sigma(n), n) > gcd(sigma(m), m) for all m < n.

The literal claim is ∀ n ≥ 1, IsMember n → IsRecord n, where IsMember is the A245786 integrality condition and IsRecord is the A216793 strict gcd-record condition.

Motivation

This first-tier OEIS conjecture was printed in 2014 and remains a universal subsequence claim linking an integrality condition to record values of gcd(sigma(n), n). A certified counterexample resolves the literal conjecture without proposing a corrected statement.

Gap

Preregistration issue #7517 and its probe report record searches dated September 13, 2026. OEIS history revisions #1 through #16 only restate the sequence and conjecture; no proof, refutation, or withdrawal appears there. Exact searches returned 0 results on arXiv and 0 on MathOverflow. OpenAlex returned 0 once and was unavailable on the repeat attempt. GitHub was rate-limited, so that surface is ASSUMED-UNVERIFIED. The sibling sequence A245778 was refuted by Max Alekseyev on July 28, 2026, but that result has a different premise and does not settle this claim. No resolution was found in the bounded surfaces that could be checked. This does not assert exhaustive literature coverage or publication priority.

Route

Let N = 275890944 = 2⁸·3·7·19·37·73. Multiplicativity gives tau(N) = 288 and sigma(N) = 919636480, hence N/288 + 919636480/N = 957958. Thus N is a member of A245786, and gcd(sigma(N), N) = 91963648.

For the smaller number M = 142990848 = 2⁹·3²·7·11·13·31, multiplicativity gives sigma(M) = 571963392 = 4M, so gcd(sigma(M), M) = M. Therefore M < N and gcd(sigma(M), M) = 142990848 > 91963648 = gcd(sigma(N), N). Consequently N is a member but not a record, refuting the subsequence conjecture.

Falsifier

A proof that every positive A245786 member satisfies the A216793 strict record condition would falsify this refutation. The kernel-checked instance at N = 275890944 and its smaller witness M = 142990848 prove the opposite, so such a proof would contradict the formal result.

Evidence

• Lean module: D5/S0/Certificates/KrizekIntegralityGcdRecordRefutation.lean.
• Main theorem: result : ¬ claim, with exactly the std3 axioms propext, Classical.choice, and Quot.sound.
• The probe recorded whole-file kernel type-checking in 19.2 seconds.
• The orchestrator independently checked the exact values for N and M with SymPy and found the first six listed members unbeaten by smaller multiply-perfect numbers.
• The probe independently found the members at most 2·10⁵ to be exactly {1, 672, 4680, 30240}, all record points, and checked the factorizations, divisor counts, divisor sums, gcd values, and rational value used here.

The bounded searches are supporting evidence only. The formal result uses the single member N = 275890944 and the smaller witness M = 142990848.

Triage

theorem. The certified member at N = 275890944 is not a gcd record, so it refutes the literal subsequence conjecture. It asserts no corrected statement and no claim about any other input.

ASSUMED-UNVERIFIED

The repeat OpenAlex search was unavailable, and the GitHub search was rate-limited, so those surfaces were not fully verified. The literature search is bounded and does not establish exhaustive coverage or publication priority. The counterexample N = 275890944 is not claimed to be the least counterexample.