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DIVISIBILITY OF CERTAIN SINGULAR OVERPARTITIONS BY POWERS OF $\\textbf{2}$ AND $\\textbf{3}
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Journal Article
DIVISIBILITY OF CERTAIN SINGULAR OVERPARTITIONS BY POWERS OF $\\textbf{2}$ AND $\\textbf{3}
2021
Overview
Andrews introduced the partition function
$\\overline {C}_{k, i}(n)$
, called the singular overpartition function, which counts the number of overpartitions of n in which no part is divisible by k and only parts
$\\equiv \\pm i\\pmod {k}$
may be overlined. We prove that
$\\overline {C}_{6, 2}(n)$
is almost always divisible by
$2^k$
for any positive integer k. We also prove that
$\\overline {C}_{6, 2}(n)$
and
$\\overline {C}_{12, 4}(n)$
are almost always divisible by
$3^k$
. Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of
$2$
satisfied by
$\\overline {C}_{6, 2}(n)$
.
Publisher
Cambridge University Press
Subject
