bibkey: layman2011a196226 authors: John W. Layman year: 2011 title: “OEIS A196226, divisor-sum residue and divisor-power congruence conjectures” doi: null url: https://oeis.org/A196226 claim: “If m is a sufficiently large term of A196226, then sigma(k,m) is congruent to 2^k + 1 + m/2 modulo m; the entry states the cases k = 2, 3, and 4 with thresholds 14, 22, and 38.” strata_touched:

• D5/S0/Certificates/LaymanDivisorPowerCongruenceRefutation license: citation-only triage: anchor

OEIS A196226

John W. Layman’s entry defines A196226 as the natural numbers m for which the sum of divisors of m, reduced modulo m, is 3 + m/2. Its conjecture comment states three universal congruences: for terms at least 14, 22, and 38, the sums of the second, third, and fourth powers of the divisors are congruent modulo m to 5 + m/2, 9 + m/2, and 17 + m/2, respectively.

The formal result uses m = 690, the first listed non-semiprime term above all three thresholds, to refute each of the three congruences.

Verified locator

• URL: https://oeis.org/A196226