Arithmetic of the 13-regular partition function modulo 3

Abstract

Let b 13 (n) denote the number of 13-regular partitions of n. We study in this paper the behavior of b 13 (n) modulo 3 where n ≡ 1 (mod 3). In particular, we identify an infinite family of arithmetic progressions modulo arbitrary powers of 3 such that b 13 (n) ≡ 0 (mod 3).

Partition identities arising from Ramanujan’s formulas for multipliers

“…For example, Hirschhorn and Sellers [21], Furcy and Penniston [17], Webb [34] and Lovejoy and Penniston [29]. Recently, Cui and Gu [16], Keith [22], and Baruah and Das [5] found several arithmetic properties and infinite families of congruences for some k-regular partitions.…”

Congruences for ℓ-regular partition functions modulo 3

“…Remark 1 The = 4 and = 13 cases of Theorem 1 are proven in [3] and [12], respectively (in fact, the congruences for b 4 (n) are shown to hold modulo 6). We include them here for completeness and note that our method, a modification of that utilized in [12], gives a uniform approach to proving all of these congruences.…”

“…In particular, Andrews, Hirschhorn, and Sellers [3] proved that b 4 (n) satisfies two infinite families of Ramanujan-type congruences modulo 3, and Webb [12] proved an analogous result for b 13 (n). Our main theorem extends these results to other values of which are congruent to 1 modulo 3.…”

“…We include them here for completeness and note that our method, a modification of that utilized in [12], gives a uniform approach to proving all of these congruences.…”

Congruences for 9-regular partitions modulo 3

“…Recently, arithmetic properties of -regular partition functions have received a great deal of attention (see, for example, [2,[4][5][6][7]10,12,[14][15][16]18,19]). Xia and Yao [18] obtained that for all integers n ≥ 0 and k ≥ 0, b 9 2 6k+7 n + 2 6k+6 − 1 3 ≡ 0 (mod 2).…”