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 $a \in A$ 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.