Divisibility of Andrews’ singular overpartitions by powers of 2 and 3

Andrews introduced the partition function C ¯ k , i ( n ) , called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ± i ( mod k ) may be overlined. He also proved that C ¯ 3 , 1 ( 9 n + 3 ) and C ¯ 3 , 1 ( 9 n + 6 ) are divisible by 3 for n ≥ 0 . Recently Aricheta proved that for an infinite family of k , C ¯ 3 k , k ( n ) is almost always even. In this paper, we prove that for any positive integer k , C ¯ 3 , 1 ( n ) is almost always divisible by 2 k and 3 k .