Ramanujan-type congruences for overpartitions modulo 16

Abstract

Abstract. Let p(n) denote the number of overpartitions of n. Hirschhorn and Sellers showed that p(4n + 3) ≡ 0 (mod 8) for n ≥ 0. They also conjectured that p(40n + 35) ≡ 0 (mod 40) for n ≥ 0. Chen and Xia proved this conjecture by using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams. In this paper, we show that p(5n) ≡ (−1) n p(4 · 5n) (mod 5) for n ≥ 0 and p(n) ≡ (−1) n p(4n) (mod 8) for n ≥ 0 by using the relation of the generating function of p(5n) modulo 5 found by Treneer and the 2-adic expansion of the generating function of p(n) due to Mahlburg. As a consequence, we deduce that p(4 k (40n + 35)) ≡ 0 (mod 40) for n, k ≥ 0. Furthermore, applying the Hecke operator on φ(q) 3 and the fact that φ(q) 3 is a Hecke eigenform, we obtain an infinite family of congrences p(4 k · 5ℓ 2 n) ≡ 0 (mod 5), where k ≥ 0 and ℓ is a prime such that ℓ ≡ 3 (mod 5) and −n ℓ = −1. Moreover, we show that p(5 2 n) ≡ p(5 4 n) (mod 5) for n ≥ 0. So we are led to the congruences p 4 k 5 2i+3 (5n ± 1) ≡ 0 (mod 5) for n, k, i ≥ 0. In this way, we obtain various Ramanujan-type congruences for p(n) modulo 5 such as p(45(3n + 1)) ≡ 0 (mod 5) and p(125(5n ± 1)) ≡ 0 (mod 5) for n ≥ 0.

Restricted k-color partitions

“…The fourth clause follows from c 2,1 (16n + 14) ≡ 0 (mod 16), which was established in [5]. The last clause follows from Theorem 3.…”

“…c 2k,1 (p(pn + r)) ≡ 0 (mod 8) (r not a quadratic residue mod p) (We note that the congruences for c 2,1 have independently appeared in preprint during the writing of this paper, proved by dissection of generating functions: [5]. They also establish c 2,1 (16n + 14) ≡ 0 (mod 16), which with this technique would require information on ν 3 , and other congruences for overpartitions modulo powers of 2.…”

New jatropholane-type diterpenes from Jatropha curcas cv. Multiflorum CY Yang

“…multiflorum was repeatedly chromatographed on silica gel, RP-18, and HPLC to yield ten jatropholane-type diterpenes (1-10). Of them, seven known compounds were identified as sikkimenoid B (4) 15 , jatrophaldehyde (5) 16 , epi-jatrophaldehyde (6) 16 , epi-jatrophol (7) 16 , jatrophol (8) 17 , jatropholone A (9)…”

Partitions into a small number of part sizes

“…The case (A, B) = (196, 70) is separate and follows immediately from Theorem 1.2 of Chen et. al in [2], which holds that p(7n) ≡ 0 (mod 16) unless 7|n.…”