Let $\mathrm{ pod}(n)$ denote the number of partitions of $n$ with odd parts distinct, and ${ { r}_{ k}}(n)$ be the number of representations of $n$ as sum of $k$ squares. We find the following two arithmetic relations: for any integer $n\ge 0$, \[\mathrm{ pod}(3n+2)\equiv 2{ { (-1)}^{ n+1}}{ { r}_{ 5}}(8n+5) \pmod{ 9}, \] and \[\mathrm{ pod}(5n+2)\equiv 2{ { (-1)}^{ n}}{ { r}_{ 3}}(8n+3) \pmod{ 5}.\] From which we deduce many interesting congruences including the following two infinite families of Ramanujan-type congruences: for $a \in \{ 11, 19\}$ and any integers $\alpha \ge 1$ and $n \ge 0$, we have \[\mathrm{ pod}\Big({ { 5}^{ 2\alpha +2}}n+\frac{ a \cdot { { 5}^{ 2\alpha +1}}+1}{ 8}\Big)\equiv 0 \pmod{ 5}.\]
Categories: math.NT