2024/04/30 by Talukdar, Pranjal
#05A17 #11F11 #11P83 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2404.19731
An integer partition of a positive integer n is called to be t-core if none of its hook lengths are divisible by t. Recently, Gireesh, Ray and Shivashankar [`A new analogue of t-core partitions', Acta Arith. 199 (2021), 33-53] introduced an analogue at(n) of the t-core partition function ct(n). They obtained certain multiplicative formulas and arithmetic identities for at(n) where t ∈ \3,4,5,8\ and studied the arithmetic density of at(n) modulo pij where t=p1a1⋯ pmam and pi≥ 5 are primes. Very recently, Bandyopadhyay and Baruah [`Arithmetic identities for some analogs of the 5-core partition function', J. Integer Seq. 27 (2024), # 24.4.5] proved new arithmetic identities satisfied by a5(n). In this article, we study the arithmetic densities of at(n) modulo arbitrary powers of 2 and 3 for t=3αm where gcd(m,6)=1. Also, employing a result of Ono and Taguchi on the nilpotency of Hecke operators, we prove an infinite family of congruences for a3(n) modulo arbitrary powers of 2.