Divisibility by 2 and 3 of certain Stirling numbers

TL;DR

This paper characterizes when certain Stirling number-based functions reach equality in a known inequality for primes 2 and 3, linking the property to the base-p digit sum condition of integers.

Contribution

It precisely determines the set of integers n for which equality holds in the inequality involving e_p for p=2 and 3, based on base-p digit sum conditions.

Findings

Equality holds when the sum of two consecutive digits in n's base-p expansion is less than p.

The characterization applies specifically for primes p=2 and 3.

The results connect digit sum properties to divisibility properties of Stirling numbers.

Abstract

The numbers e_p(k,n) defined as min(nu_p(S(k,j)j!): j >= n) appear frequently in algebraic topology. Here S(k,j) is the Stirling number of the second kind, and nu_p(-) the exponent of p. The author and Sun proved that if L is sufficiently large, then e_p((p-1)p^L + n -1, n) >= n-1+nu_p([n/p]!). In this paper, we determine the set of integers n for which equality holds in this inequality when p=2 and 3. The condition is roughly that, in the base-p expansion of n, the sum of two consecutive digits must always be less than p.