bibkey: kagey2021a343825
authors: P. Kagey; OEIS Foundation Inc.
year: 2021
title: "OEIS A343825, prime-power sequence minima and A071537"
doi: null
url: https://oeis.org/A343825
claim: "Conjecture: T(p,k) = A071537(k) for fixed k and sufficiently large prime p."
strata_touched:
  - D5/S3/Arith/Factorization/OeisA343825KageyPowerSequences
license: citation-only
triage: anchor

Prime-power sequence minima

Verified locator

DOI: null

Source: https://oeis.org/A343825

The entry is unpaginated. Revision 22, NAME:

Table read by antidiagonals upward: T(n,k) is the least m such that there exists a sequence k = b_1 <= b_2 <= … <= b_t = m such that no term appears n or more times and the product of the sequence is of the form c^n, where c is an integer; n >= 1 and k >= 0.

Revision 22, COMMENT:

Conjecture: T(p,k) = A071537(k) for fixed k and sufficiently large prime p.

The linked entry A071537, NAME:

Least integer m >= n such that there exist integers k_n, k_n+1, …, k_m with nonzero k_n, for which n^(k_n) * (n+1)^(k_n+1) * … * m^(k_m) = 1.

A071537 has offset 1. The assertion therefore quantifies over k ≥ 1. Integer exponents are interpreted in the rationals. The least endpoint is extended by zero if its defining set is empty.

The prime-row permutation assertion is a separate question. The eventual-equality proof does not settle it, characterize exceptional primes, or give an explicit threshold as a function of k.

The source and literature screen are specified in the problem source. Completeness of that literature screen is ASSUMED-UNVERIFIED.