Some Notes on Partitions Inspired by Schur

Abstract

Let $f_6(n)$ denote the number of partitions of the natural number $n$ into parts co-prime to $6$. This function was originally studied by Schur. We derive two explicit formulas for $f_6(n)$, one of them in terms of the partition function $p(n)$. We also derive three recurrences for $f_6(n)$.