Divisibility of sums of some restricted partition numbers by \({\textbf {2}}\), \({\textbf {3}}\) and \({\textbf {4}}\)

In this paper, we show that sums of partition numbers into distinct odd parts, sums of 2-core partition numbers and sums of partition numbers into odd parts are divisible by 2, 3 and 4. For example, if \(p_{od}(n)\) $p_{\mathit{od}}(n)$ denotes the number of partitions into distinct odd parts of a positive integer n, then for non-negative integer s and \(\omega (k)=k(3k+1)/2\) $\omega (k)=k(3k+1)/2$ : \(\begin{aligned} \sum _{k=0}^{\infty }p_{od}(48s+46-\omega (-2k))+\sum _{k=1}^{\infty }p_{od}(48s+46-\omega (2k))\equiv 0\pmod {4}. \end{aligned}\) $$\begin{array}{c}\hfill \sum _{k=0}^{\infty }{p}_{\mathit{od}}(48s+46-\omega (-2k))+\sum _{k=1}^{\infty }{p}_{\mathit{od}}(48s+46-\omega (2k))\equiv 0(mod4).\end{array}$$