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

Abstract

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 \(a \in A \) and \(b \in B \), in an even number of ways. Furthermore, the generating functions for \(A \) and \(B \) over \(\FF_2 \) are reciprocals in \(\FF_2 [[q]] \). 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 \(k(3k\pm1)/2 \), 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 \(1/2 \). 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 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.