Some Notes on Partitions Inspired by Schur

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 \( f_6(n) \).