bibkey: schulte2025a000680 authors: Werner Schulte year: 2025 title: “OEIS A000680, (2n)!/2^n, with the primality criterion (2n+1) | a(n) + 2^n iff 2n+1 is prime” doi: null url: https://oeis.org/A000680 claim: “%N a(n) = (2n)!/2^n. %C Conjecture: For n > 0 (a(n) + 2^n) is a multiple of (2n + 1) if and only if (2n + 1) is a prime number. - Werner Schulte, Oct 05 2025” strata_touched:

• D5/S3/Arith/Congruence/SchulteHalfFactorialPrimeCriterion license: citation-only triage: anchor

OEIS A000680

Werner Schulte’s 2025 comment proposes the half-factorial primality criterion for the sequence A000680.

Verified locator

• URL: https://oeis.org/A000680

%N a(n) = (2n)!/2^n.
%O 0,3
%A _N. J. A. Sloane_
%C Conjecture: For n > 0 (a(n) + 2^n) is a multiple of (2*n + 1) if and only if (2*n + 1) is a prime number. - _Werner Schulte_, Oct 05 2025

Readings of 2026-09-16: the OEIS entry still marks the line Conjecture (added 2025-10-05) with no proof line; OpenAlex "A000680" returns 33 hits, none about this criterion; Math.SE API 24 hits, none about it; arXiv was not searched (no API response). Wilson’s theorem and the odd-composite factorial divisibility are textbook facts; the named criterion itself was not found in these surfaces. Historical openness beyond them is ASSUMED-UNVERIFIED.