Congruences for the Number of Cubic Partitions Derived from Modular Forms

Abstract

We obtain congruences for the number a(n) of cubic partitions using modular forms. The notion of cubic partitions is introduced by Chan and named by Kim in connection with Ramanujan's cubic continued fractions. Chan has shown that a(n) has several analogous properties to the number p(n) of partitions, including the generating function, the continued fraction, and congruence relations. To be more specific, we show that $a(25n+22)\equiv 0(\mathrm{mod}5)$, $a(49n+15)\equiv a(49n+29)\equiv a(49n+36)\equiv a(49n+43)\equiv 0(\mathrm{mod}7)$. Furthermore, we prove that a(n) takes infinitely many even values and infinitely odd values.

15 pages

References in corpus (1)

• Ramanujan congruences for a class of eta quotients

Cited by in corpus (7)

• The number of cubic partitions modulo powers of 5
• New congruences for 2-color partitions
• Multiranks and classical theta functions
• Arithmetic properties for cubic partition pairs modulo powers of 3
• Ramanujan-type congruences for $2$-color partition triples
• Ramanujan-Type congruences for cubic partition functions
• Congruences for an arithmetic function from 3-colored Frobenius partitions