Ramanujan-type congruences for overpartitions modulo 16

Let (bar{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 (bar{p}(n)) and derived a number of congruences for (bar{p}(n)) modulo 4, 8 and 64 including (bar{p}(8n+7)equiv 0 pmod {64}) for (nge 0). In this paper, we give a 16-dissection of the generating function for (bar{p}(n)) modulo 16 and show that (bar{p}(16n+14)equiv 0pmod {16}) for (nge 0). Moreover, using the (2)-adic expansion of the generating function for (bar{p}(n)) according to Mahlburg, we obtain that (bar{p}(ell ^2n+rell )equiv 0pmod {16}), where (nge 0), (ell equiv -1pmod {8}) is an odd prime and (r) is a positive integer with (ell not mid r). In particular, for (ell =7) and (nge 0), we get (bar{p}(49n+7)equiv 0pmod {16}) and (bar{p}(49n+14)equiv 0pmod {16}). We also find four congruence relations: (bar{p}(4n)equiv (-1)^nbar{p}(n) pmod {16}) for (nge 0), (bar{p}(4n)equiv (-1)^nbar{p}(n)pmod {32}) where (n) is not a square of an odd positive integer, (bar{p}(4n)equiv (-1)^nbar{p}(n)pmod {64}) for (nnot equiv 1,2,5pmod {8}) and (bar{p}(4n)equiv (-1)^nbar{p}(n)pmod {128}) for (nequiv 0pmod {4}).