Congruences via Partitions with Exactly Two Part Sizes

We prove the congruence ${\sum }_{1\le k<\sqrt{N}}{\sigma }_0(N-k^2)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$, where ${\sigma }_0(m)$ denotes the number of positive divisors of $m$, for $N=An+B$ with $(A,B)\in \{(16,14),$ $(36,30),$ $(72,42),$ $(196,70),$ $(252,114)\}$. Our proof relies on a result of Keith which states that ${\nu }_2(N)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$, where ${\nu }_2(N)$ is the number of partitions of $N$ with exactly two part sizes. Inspired by Dewitt and Keith, our approach combines combinatorial arguments with modular arithmetic techniques.