Abstract: 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 M i for which p ( M i ) 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 ).