On the parity of generalized partition functions III

Abstract

Improving on some results of J.-L. Nicolas \cite {Ndeb}, the elements of the set $\mathcal{A}=\mathcal{A}(1+z+z^3+z^4+z^5)$, for which the partition function $p(\mathcal{A},n)$ (i.e. the number of partitions of $n$ with parts in $\mathcal{A}$) is even for all $n\ge 6$ are determined. An asymptotic estimate to the counting function of this set is also given.