Abstract
Abstract
An integer partition of a positive integer n is called t-core if none of its hook lengths is divisible by t. Gireesh et al. [‘A new analogue of t-core partitions’, Acta Arith.199 (2021), 33–53] introduced an analogue
$\overline {a}_t(n)$
of the t-core partition function. They obtained multiplicative formulae and arithmetic identities for
$\overline {a}_t(n)$
where
$t \in \{3,4,5,8\}$
and studied the arithmetic density of
$\overline {a}_t(n)$
modulo
$p_i^{\,j}$
where
$t=p_1^{a_1}\cdots p_m^{a_m}$
and
$p_i\geq 5$
are primes. Bandyopadhyay and Baruah [‘Arithmetic identities for some analogs of the 5-core partition function’, J. Integer Seq.27 (2024), Article no. 24.4.5] proved new arithmetic identities satisfied by
$\overline {a}_5(n)$
. We study the arithmetic densities of
$\overline {a}_t(n)$
modulo arbitrary powers of 2 and 3 for
$t=3^\alpha m$
where
$\gcd (m,6)$
=1. Also, employing a result of Ono and Taguchi [‘2-adic properties of certain modular forms and their applications to arithmetic functions’, Int. J. Number Theory1 (2005), 75–101] on the nilpotency of Hecke operators, we prove an infinite family of congruences for
$\overline {a}_3(n)$
modulo arbitrary powers of 2.
Publisher
Cambridge University Press (CUP)