On the Reciprocal of the Binary Generating Function for the Sum of Divisors

If A is a set of natural numbers containing 0, then there is a unique nonempty “reciprocal” set B of natural numbers (containing 0) such that every positive integer can be written in the form a+ b, where a ∈ A and b ∈ B, in an even number of ways. Furthermore, the generating functions for A and B over F2 are reciprocals in F2[[q]]. We consider the reciprocal set B for the set A containing 0 and all integers such that σ(n) is odd, where σ(n) is the sum of all the positive divisors of n. This problem is motivated by Euler’s pentagonal number theorem, a corollary of which is that the set of natural numbers n so that the number p(n) of partitions of an integer n is odd is the reciprocal of the set of generalized pentagonal numbers (integers of the form k(3k ± 1)/2, where k is a natural number). An old (1967) conjecture of Parkin and Shanks is that the density of integers n so that p(n) is odd (equivalently, even) is 12 . Euler also found that σ(n) satisfies an almost identical recurrence as that given by the pentagonal number theorem, so we hope to shed light on the Parkin-Shanks conjecture by computing the density of the reciprocal of the set containing the natural numbers 1 with σ(n) odd (σ(0) = 1 by convention). We conjecture this particular density is 1 32 and prove that it lies between 0 and 1 16 . We finish with a few surprising connections between certain Beatty sequences and the sequence of integers n for which σ(n) is odd.