A short Proof of a conjecture by Hirschhorn and Sellers on Overpartitions

Let $\bar{p}(n)$ be the number of overpartitions of $n$, we establish and give a short elementary proof of the following congruence $$\bar{p}(4^{\alpha }(40n+35))\equiv 0(40),$$ where $\alpha ,n$ are nonnegative integers. By letting $\alpha =0$ we proved a conjecture of Hirschhorn and Sellers. Some new congruences for $\bar{p}(n)$ modulo 3 and 5 have also been found, including the following two infinite families of Ramanujan-type congruences: for any integers $n\ge 0$ and $\alpha \ge 1$, $$\bar{p}(5^{2\alpha +1}(5n+1))\equiv \bar{p}(5^{2\alpha +1}(5n+4))\equiv 0(5)\mathrm{.}$$