The OEIS A231548 Antisigma Gap-Three Conjecture
Abstract
At n = 332640, antisigma decreases across a gap of three.
Definition 1.1 (The antisigma function).
Formalization. D5/S0/Certificates/KrizekAntisigmaDecreaseRefutation.antisigma (✓ std3).
Citation. Jaroslav Krizek (2013). OEIS A231548, Numbers n such that 2n - 1 < sigma(n) - sigma(n-2)*. URL: https://oeis.org/A231548.
Commentary.
For each natural n, the finite interval contains the integers from one through n. The filter retains exactly those d that do not divide n, and the outer sum adds the retained values.
Definition 1.2 (Krizek’s gap-three conjecture).
Formalization. D5/S0/Certificates/KrizekAntisigmaDecreaseRefutation.claim (✓ std3).
Citation. Jaroslav Krizek (2013). OEIS A231548, Numbers n such that 2n - 1 < sigma(n) - sigma(n-2)*. URL: https://oeis.org/A231548.
Commentary.
For every natural n at least three, the conjecture says that antisigma at n minus three is at most antisigma at n.
Theorem 1.3 (The conjecture fails at n = 332640).
Proof. Machine-checked in Lean as D5/S0/Certificates/KrizekAntisigmaDecreaseRefutation.result (✓ std3). ∎
Resolves. Problems/oeis-a231548-antisigma-decrease-refutation (refuted) by D5/S0/Certificates/KrizekAntisigmaDecreaseRefutation.result.
Source. Repository-derived.
Acknowledgement. Jaroslav Krizek (2013). OEIS A231548, Numbers n such that 2n - 1 < sigma(n) - sigma(n-2)*. URL: https://oeis.org/A231548.
Commentary.
At n = 332640, the divisor sums are sigma(332640) = 1451520 and sigma(332637) = 443520. The complement identity gives antisigma(332640) = 55323399600 and antisigma(332637) = 55323409683. The first value is smaller, so the universal gap-three inequality is false.