Congruences via Partitions with Exactly Two Part Sizes

Sittinon Jirattikansakul, Teeradej Kittipassorn, Kraiwich Kongsiri +2

We prove the congruence ${\sum }_{1\le k<\sqrt{N}}\ddot{I}{ƒ}_0(N-k^2)\equiv 0(\mathrm{mod}4)$, where $\ddot{I}{ƒ}_0(m)$ denotes the number of positive divisors of $m$, for $N=An+B$ with $(A,B) \…