bibkey: bala2022a122399 authors: Peter Bala year: 2022 title: “OEIS A122399 (Jovovic 2006), conjecture: a(n) mod p is purely periodic with period p - 1” doi: null url: https://oeis.org/A122399 claim: “a(n) = Sum_{k=0..n} k^n * k! * Stirling2(n,k). Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1.” strata_touched:

• D5/S1/Recurrence/Parity/StirlingPowerFactorialPrimePeriod license: citation-only triage: anchor

OEIS A122399

Vladeta Jovovic introduced the entry on August 31, 2006. Peter Bala’s comment of May 31, 2022 states the conjecture quoted above. The conjecture is Tier 1 by its 2022 date, independently of the entry’s 2006 creation date. The cited entry presents it as a conjecture without a proof note.

The sequence has offset zero and is defined by the finite sum in the claim. The formalization uses Mathlib’s Nat.stirlingSecond, proves its factorial-weighted inclusion-exclusion formula, and reduces the sum modulo each prime to the fixed window 0 <= k < p. The resulting double sum is a linear combination of powers (k*j)^n; Fermat’s little theorem makes each term periodic for positive n, with zero bases handled separately.

The result asserts that p - 1 is a period and that every nonnegative multiple of it preserves every positive-index term. It does not assert minimality of that period. The window identity also holds at index zero.

Verified locator

• URL: https://oeis.org/A122399