bibkey: kagey2021a343881
authors: P. Kagey; OEIS Foundation Inc.
year: 2021
title: "OEIS A343881, power-product minima and A064549"
doi: null
url: https://oeis.org/A343881
claim: "Conjecture: T(p,k) = A064549(k) for fixed k > 1 and sufficiently large p."
strata_touched:
  - D5/S3/Arith/Factorization/OeisA343881KageyPowerProducts
license: citation-only
triage: anchor

Power-product minima

Verified locator

DOI: null

Source: https://oeis.org/A343881

The entry is unpaginated. Revision 16, NAME:

Table read by antidiagonals upward: T(n,k) is the least integer m > k such that k^x * m^y = c^n for some positive integers c, x, and y where x < n and y < n; n >= 2, k >= 1.

Revision 16, COMMENT:

Conjecture: T(p,k) = A064549(k) for fixed k > 1 and sufficiently large p.

The linked A064549, revision 71, NAME:

a(n) = n * Product_{primes p|n} p.

The first definition uses natural numbers and requires positive c, x and y. The minimum is extended by zero when its defining set is empty. The conjecture quantifies over primes beyond a threshold depending on k.

At k=12 the proposed value is 72. Prime coordinates of 12 and 72 force the exponent y to be divisible by n, contrary to 0<y<n. A working larger factor exists for every prime exponent, so this obstructs the proposed minimum at every prime. The universal corrected minimum 144 for primes at least 7 is not established by this formalization.

The source’s separate COMMENT, “For prime p, the p-th row consists of distinct integers.”, and the equal-prime-exponent restriction of the conjecture remain unresolved here.

The source and literature screen are specified in the problem source. Literature completeness is ASSUMED-UNVERIFIED.