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