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 Euler’s 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). A partition of a non-negative integer n is any non-increasing sequence of positive integers whose sum is n, and let p(n) denote the number of partitions of n. Euler’s generating function for p(n) is given by the infinite product: