Arithmetic properties of 5-regular partitions into distinct parts

A partition is said to be [Formula: see text]-regular if none of its parts is a multiple of [Formula: see text]. Let [Formula: see text] denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of n. This function has also close connections to representation theory and combinatorics. In this paper, we study arithmetic properties of [Formula: see text]. We provide full characterization of the parity of [Formula: see text], present several congruences modulo 4, and prove that the generating function of the sequence [Formula: see text] is lacunary modulo any arbitrary positive powers of 5.