New Congruences of Partitions With Odd Parts Distinct

الأصل · EN

Let pod(n) denote the number of partitions of n with odd parts distinct, and rₖ(n) be the number of representations of n as sum of k squares. We find the following two arithmetic relations: for any integer n≥ 0, pod(3n+2)≡ 2(-1)ⁿ⁺¹r₅(8n+5) 9, and pod(5n+2)≡ 2(-1)ⁿr₃(8n+3) 5. From which we deduce many interesting congruences including the following two infinite families of Ramanujan-type congruences: for a ∈ {11, 19} and any integers α≥ 1 and n ≥ 0, we have pod(5²α⁺²n+a · 5²α⁺¹+18)≡ 0 5.