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