Abstract. In this article, we consider various arithmetic properties of the function p o (n) which denotes the number of overpartitions of n using only odd parts. This function has arisen in a number of recent papers, but in contexts which are very different from overpartitions. We prove a number of arithmetic results including several Ramanujan-like congruences satisfied by p o (n) and some easily-stated characterizations of p o (n) modulo small powers of two. For example, it is proven that, for n ≥ 1, p o (n) ≡ 0 (mod 4) if and only if n is neither a square nor twice a square.
Formulas for the number of primitive representations of any integer n as a sum of k squares are given, for 2 ≤ k ≤ 8, and for certain values of n, for 9 ≤ k ≤ 12. The formulas have a similar structure and are striking for their simplicity.
There are three modular forms a(q), b(q), c(q) involved in the parametrization of the hypergeometric function analogous to the classical θ2(q), θ3(q), θ4(q) and the hypergeometric function We give elliptic function generalizations of a(q), b(q), c(q) analogous to the classical theta-function θ(z, q). A number of identities are proved. The proofs are self-contained, relying on nothing more than the Jacobi triple product identity
In this work, we consider the function ped(n), the number of partitions of an integer n wherein the even parts are distinct (and the odd parts are unrestricted). Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan's congruences for the unrestricted partition function p(n). We prove a number of results for ped(n) including the following: For all n ≥ 0, ped(9n + 4) ≡ 0 (mod 4) and ped(9n + 7) ≡ 0 (mod 12).Indeed, we compute appropriate generating functions from which we deduce these congruences and find, in particular, the surprising result thatWe also show that ped(n) is divisible by 6 at least 1/6 of the time.
In a recent paper, Calkin et al. [N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston and J. Radder, 'Divisibility properties of the 5-regular and 13-regular partition functions ', Integers 8 (2008), #A60] used the theory of modular forms to examine 5-regular partitions modulo 2 and 13-regular partitions modulo 2 and 3; they obtained and conjectured various results. In this note, we use nothing more than Jacobi's triple product identity to obtain results for 5-regular partitions that are stronger than those obtained by Calkin and his collaborators. We find infinitely many Ramanujan-type congruences for b 5 (n), and we prove the striking result that the number of 5-regular partitions of the number n is even for at least 75% of the positive integers n.
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