Ramanujan-type Congruences for Overpartitions Modulo 16

Abstract

Let $\overline{p}(n)$ denote the number of overpartitions of $n$. Recently, Fortin-Jacob-Mathieu and Hirschhorn-Sellers independently obtained 2-, 3- and 4-dissections of the generating function for $\overline{p}(n)$ and derived a number of congruences for $\overline{p}(n)$ modulo $4$, $8$ and $64$ including $\overline{p}(5n+2)\equiv 0(\mathrm{mod}4)$, $\overline{p}(4n+3)\equiv 0(\mathrm{mod}8)$ and $\overline{p}(8n+7)\equiv 0(\mathrm{mod}64)$. By employing dissection techniques, Yao and Xia obtained congruences for $\overline{p}(n)$ modulo $8,16$ and $32$, such as $\overline{p}(48n+26)\equiv 0(\mathrm{mod}8)$, $\overline{p}(24n+17)\equiv 0(\mathrm{mod}16)$ and $\overline{p}(72n+69)\equiv 0(\mathrm{mod}32)$. In this paper, we give a 16-dissection of the generating function for $\overline{p}(n)$ modulo 16 and we show that $\overline{p}(16n+14)\equiv 0(\mathrm{mod}16)$ for $n\ge 0$. Moreover, by using the $2$-adic expansion of the generating function of $\overline{p}(n)$ due to Mahlburg, we obtain that $\overline{p}({\ell }^{2}n+r\ell )\equiv 0(\mathrm{mod}16)$, where $n\ge 0$, $\ell \equiv -1(\mathrm{mod}8)$ is an odd prime and $r$ is a positive integer with $\ell \nmid r$. In particular, for $\ell =7$, we get $\overline{p}(49n+7)\equiv 0(\mathrm{mod}16)$ and $\overline{p}(49n+14)\equiv 0(\mathrm{mod}16)$ for $n\ge 0$. We also find four congruence relations: $\overline{p}(4n)\equiv (-1{)}^n\overline{p}(n)(\mathrm{mod}16)$ for $n\ge 0$, $\overline{p}(4n)\equiv (-1{)}^n\overline{p}(n)(\mathrm{mod}32)$ for $n$ being not a square of an odd positive integer, $\overline{p}(4n)\equiv (-1{)}^n\overline{p}(n)(\mathrm{mod}64)$ for $n\not\equiv 1,2,5(\mathrm{mod}8)$ and $\overline{p}(4n)\equiv (-1{)}^n\overline{p}(n)(\mathrm{mod}128)$ for $n\equiv 0(\mathrm{mod}4)$.

12 pages