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 (\sigma(n) ) is odd, where (\sigma(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 (\sigma(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 (\sigma(n) ) odd ((\sigma(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 (\sigma(n) ) is odd.