Congruences modulo 8 for $$(2,\, k)$$ (2,k) -regular overpartitions for odd $$k > 1$$ k>1

Abstract

Abstract In this paper, we study various arithmetic properties of the function $$\overline{p}_{2,\,\, k}(n)$$ p¯2,k(n) , which denotes the number of $$(2,\,\, k)$$ (2,k) -regular overpartitions of n with odd $$k > 1$$ k>1 . We prove several infinite families of congruences modulo 8 for $$\overline{p}_{2,\,\, k}(n)$$ p¯2,k(n) . For example, we find that for all non-negative integers $$\beta , n$$ β,n and $$k\equiv 1\pmod {8}$$ k≡1(mod8) , $$\overline{p}_{2,\,\, k}(2^{1+\beta }(16n+14))\equiv ~0\pmod {8}$$ p¯2,k(21+β(16n+14))≡0(mod8) .

Keywords

• 05A15
• 05A17
• 11P83