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Distribution of a certain partition function modulo powers of primes. (English)
Acta Math. Sin., Engl. Ser. 27, No. 4, 625-634 (2011).
Summary: In this paper, we study a certain partition function $a(n)$ defined by $Σ_{n \geq 0} a(n)q^{n}:= Π_{n=1}(1-q^{n})^{-1}(1-q^{2n})^{-1}$. We prove that given a positive integer $j \geq 1$ and a prime $m \geq 5$, there are infinitely many congruences of the type $a(An + B) \equiv 0 \, \pmod{m^{j}}$. This work is inspired by Ono’s ground breaking result in the study of the distribution of the partition function $p(n)$.