Layman’s OEIS A196226 Divisor-Power Congruence Conjectures
Abstract
The integer 690 refutes John W. Layman’s three divisor-power congruence conjectures.
Definition 1.1 (Membership in OEIS A196226).
Formalization. D5/S0/Certificates/LaymanDivisorPowerCongruenceRefutation.membership (✓ std3).
Citation. John W. Layman (2011). OEIS A196226, divisor-sum residue and divisor-power congruence conjectures. URL: https://oeis.org/A196226.
Commentary.
A natural number m is a member when it is even and the remainder of sigma sub one of m modulo m equals 3 + m/2. Here slash denotes natural-number integer division; the divisibility clause makes m/2 exact.
Definition 1.2 (Layman’s three congruence conjectures).
Formalization. D5/S0/Certificates/LaymanDivisorPowerCongruenceRefutation.claim (✓ std3).
Citation. John W. Layman (2011). OEIS A196226, divisor-sum residue and divisor-power congruence conjectures. URL: https://oeis.org/A196226.
Commentary.
The claim is the disjunction of the three published universal congruences. Their thresholds are 14, 22, and 38; their divisor-power exponents are 2, 3, and 4; and their constants are 5, 9, and 17. Nat.ModEq(m,x,y) means that x and y are congruent modulo m, and every slash denotes natural-number integer division.
Theorem 1.3 (All three conjectures are false).
Proof. Machine-checked in Lean as D5/S0/Certificates/LaymanDivisorPowerCongruenceRefutation.result (✓ std3). ∎
Resolves. Problems/oeis-a196226-layman-divisor-power-congruence-refutation (refuted) by D5/S0/Certificates/LaymanDivisorPowerCongruenceRefutation.result.
Source. Repository-derived.
Acknowledgement. John W. Layman (2011). OEIS A196226, divisor-sum residue and divisor-power congruence conjectures. URL: https://oeis.org/A196226.
Commentary.
At m = 690, sigma sub one is 1728, so the membership remainder is 348. The next three divisor-power sums are 689000, 386358336, and 244202442248. Their remainders modulo 690 are 380, 426, and 668, rather than the required 350, 354, and 362. Thus each disjunct fails.