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On the Reciprocal of the Binary Generating Function for the Sum of Divisors
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Joshua N. Cooper

Department of Mathematics

University of South Carolina

Columbia, SC 29208

USA

Alexander W. N. Riasanovsky

Department of Mathematics

University of Pennsylvania

Philadelphia, PA 19104

USA

**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* ∈ *A* and *b* ∈ *B*,
in an even number of ways.
Furthermore, the generating functions for *A* and
*B* over **F**_{2} are reciprocals in
**F**_{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 Eulerâ€™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*(3*k* ± 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.

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(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.

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