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

Joshua Cooper; Alexander Riasanovsky

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arXiv · 1409.2909

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

Joshua Cooper·Alexander Riasanovsky

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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.

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Joshua Cooper, Alexander Riasanovsky. 2014-09-09. On the Reciprocal of the Binary Generating Function for the Sum of Divisors. https://arxiv.org/abs/1409.2909

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