Divisibility of sums of some restricted partition numbers by 2, 3 and 4

In this paper, we show that sums of partition numbers into distinct odd parts, sums of2-core partition numbers and sums of partition numbers into odd parts are divisible by 2, 3 and 4. For example, if p(od)(n)denotes the number of partitions into distinct odd parts of a positive integer n, then for non-negative integersand omega(k)=k(3k+1)/2: (infinity)& sum;(k=0)p(od)(48s+46-omega(-2k))+(infinity)& sum;(k=1)p(od)(48s+46-omega(2k))equivalent to 0 (mod 4)