bibkey: murthy2004a096127 authors: Amarnath Murthy year: 2004 title: “OEIS A096127, a(n) is the largest k such that (n^2)!/(n!)^k is an integer” doi: null url: https://oeis.org/A096127 claim: “a(n) is the largest k such that (n^2)!/(n!)^k is an integer. Conjecture: a(n)=n+1 only when n is prime or a power of a prime. [Verified for n=2..5000. - Amiram Eldar, Apr 06 2021]” strata_touched:

• D5/S1/Recurrence/Invariants/FactorialSquareDivisibilityPrimePower license: citation-only triage: anchor

OEIS A096127

Amarnath Murthy’s entry defines a(n) as the largest exponent for which (n!)^a(n) divides (n^2)!, with offset 2. The comment quoted below states that equality a(n)=n+1 occurs only for primes and prime powers. OEIS A096126 records the same assertion.

The formal theorem proves the closed biconditional: the exponent n+1 works but n+2 does not exactly when n is a prime power. Legendre’s formula and base-p digit sums compare the two factorial valuations.

Verified locator

• URL: https://oeis.org/A096127
• COMMENTS: Conjecture: a(n)=n+1 only when n is prime or a power of a prime. [Verified for n=2..5000. - Amiram Eldar, Apr 06 2021]
• NAME: a(n) is the largest k such that (n^2)!/(n!)^k is an integer.