Journal of Integer Sequences, Vol. 16 (2013), Article 13.1.8

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

Joshua N. Cooper
Department of Mathematics
University of South Carolina
Columbia, SC 29208

Alexander W. N. Riasanovsky
Department of Mathematics
University of Pennsylvania
Philadelphia, PA 19104


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 aA and bB, 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 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.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequences A000203 A001952 A001954 A003151 A003152 A028982 A052002 A192628 A192717 A192718 A197878 A215247.)

Received August 10 2012; revised version received January 24 2013. Published in Journal of Integer Sequences, January 26 2013.

Return to Journal of Integer Sequences home page