Some Arithmetic Properties of Overpartition k-Tuples

Abstract

Recently, Lovejoy introduced the construct of overpartition pairs which are a natural generalization of overpartitions. Here we generalize that idea to overpartition ktuples and prove several congruences related to them. We denote the number of overpartition k-tuples of a positive integer n by p k (n) and prove, for example, that for all n ≥ 0, p t−1 (tn + r) ≡ 0 (mod t) where t is prime and r is a quadratic nonresidue mod t.

Congruences for rs(n)

“…Let r s (n) denote the number of representations of n as the sum of s squares of integers, that is, r s (n) is the number of solutions to n = x 2 1 + x 2 2 + · · · + x 2 s in integers x i . Numerous beautiful results are obtained for r s (n).…”

Arithmetic properties of $\ell$-regular overpartition pairs

“…For example, pp(3n + 2) ≡ 0 (mod 3) . For more details on arithmetic properties of overpartition pairs one can see [8,15,19]. * Correspondence: msmnaika@rediffmail.com 2010 AMS Mathematics Subject Classification: 11P83, 05A15, 05A17.…”

Arithmetic properties of overpartition triples

“…Keister, Sellers and Vary [18] showed that if k = 2 m r where m ≥ 0, and r ≥ 1 is an odd integer, then…”