bibkey: oeis2022a357512 authors: Peter Bala year: 2022 title: A357512 — fifth-weighted Apery sums and fourth-power divisibility doi: null url: https://oeis.org/search?q=id:A357512&fmt=json claim: The entry conjectures n^4 divides a(n-1) for every n congruent to 1 or 5 modulo 6. strata_touched:

• D5/S3/ArithSums/A357512PrimeDivisibility license: citation-only triage: anchor

The composite-index conjecture in A357512

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doi: null url: https://oeis.org/search?q=id:A357512&fmt=json

The entry has 17 DATA terms with offset zero and defines the sum of k^5 choose(n,k)^2 choose(n+k,k)^2 for 0 <= k <= n. Its FORMULA section conjectures fourth-power divisibility at n congruent to 1 or 5 modulo 6. The COMMENT states the broader odd-weight family and proposes the exceptional set {3} for this sequence. The author line credits Peter Bala, October 2, 2022.

Scope

The entry supplies the conjecture, not its proof. The repository theorem proves it for all odd natural n not divisible by three, including composite n. It uses an exact square-factor reduction and an integral telescoping identity for the remaining sum. Only the constants two and three are cancelled. The proof is a repository derivation; no claim of global priority is made. The other odd weights and the necessity of the exceptional set are outside the theorem.