Divisibility of certain partition functions by powers of primes (Q1127624)

Let \(k=p_1^{a_1}p_w^{a_2}\cdots p_m^{a_m}\) be the prime factorization of a positive integer \(k\), and let \(b_k(n)\) denote the number of partitions of a non-negative integer \(n\) into parts, none of which are multiples of \(k\). If \(M\) is a positive integer, let \(S_k(N;M)\) be the number of positive integers \(n\leq N\) for which \(b_k(n)\equiv 0\pmod M\). If \(p_i^{a_i}\geq\sqrt k\), the authors use the theory of modular forms to prove that, for every positive integer \(j\), \[ \lim_{N\to\infty}{\frac{S_k(N;p_i^j)}{N}=1.} \] The authors also show that almost every \(n\equiv r\pmod t\) satisfies \(b_k(n)\equiv 0\pmod {p_i^j},\) as well as proving that there are infinitely many \(n\) such that \(b_k(n)\not\equiv 0\pmod {p_i^j}\), provided that there is at least one such \(n\). They conclude by giving an upper bound for the smallest such \(n\), if it exists.