A partition of a positive integer n is said to be t-core if none of its hook lengths are divisible by t. Recently, two analogues, \({\overline{a}}_t(n)\) and \({\overline{b}}_t(n)\) , of the t-core partition function, \(c_t(n)\) , have been introduced by Gireesh et al. (Acta Arith. 199:33-53, 2021) and Bandyopadhyay and Baruah (J. Integer Seq. 27:24, 2024), respectively. In this article, we prove the lacunarity of \({\overline{b}}_t(n)\) modulo arbitrary powers of 2 and 3 for \(t=3^\alpha m\) where \(\gcd (m,6)\) =1. For a fixed positive integer k and prime numbers \(p_i\ge 5\) , we also study the arithmetic density of \({\overline{b}}_t(n)\) modulo \(p_i^k\) where \(t=p_1^{a_1}\cdots p_m^{a_m}\) . We further find an infinite family of congruences for \({\overline{b}}_3(n)\) modulo arbitrary powers of 2 by employing a result of Ono and Taguchi on the nilpotency of Hecke operators. We also study the arithmetic density of \({\overline{b}}_2(n)\) and deduce some infinite families of congruences for \({\overline{b}}_2(n)\) modulo 2 using Newman’s identity and a result of Keith and Zanello.
