Arithmetic properties of an analogue of t-core partitions

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 āₜ(n) of the t-core partition function cₜ(n). They obtained certain multiplicative formulas and arithmetic identities for āₜ(n) where t ∈ \3,4,5,8\ and studied the arithmetic density of āₜ(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 ā₅(n). In this article, we study the arithmetic densities of āₜ(n) modulo arbitrary powers of 2 and 3 for t=3^α m where (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 ā₃(n) modulo arbitrary powers of 2.