Odd values of the partition function

Let p(n) denote the number of partitions of an integer n. Recently, the author has shown that in any arithmetic progression r (mod t), there exist infinitely many N for which p(N) is even, and there are infinitely many M for which p(M) is odd, provided there is at least one such M. Here we construct finite sets of integers Mi for which p(Mi) is odd for an odd number of i. Whereas Eulerʹs recurrence allows us to find odd values of p(n) when we already have one, the methods we describe do not rely on already having an odd value of p(n).