The Ramanujan Journal

, Volume 1, Issue 1, pp 25–34

# Divisibility of Certain Partition Functions by Powers of Primes

• Basil Gordon
• Ken Ono
Article

DOI: 10.1023/A:1009711020492

Gordon, B. & Ono, K. The Ramanujan Journal (1997) 1: 25. doi:10.1023/A:1009711020492

## Abstract

Let $$k = p_1^{a_1 } p_2^{a_2 } \cdot \cdot \cdot p_m^{a_m }$$ be the prime factorization of a positive integer k and let bk(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 Sk(N; M) be the number of positive integers ≤ N for which bk(n)≡ 0(modM). If $$p_i^{a_i } \geqslant \sqrt k$$ we prove that, for every positive integer j$$\mathop {\lim }\limits_{N \to \infty } \frac{{S_k (N;p_i^j )}} {N} = 1.$$ In other words for every positive integer j,bk(n) is a multiple of $$p_i^j$$ for almost every non-negative integer n. In the special case when k=p is prime, then in representation-theoretic terms this means that the number ofp -modular irreducible representations of almost every symmetric groupSn is a multiple of pj. We also examine the behavior of bk(n) (mod $$p_i^j$$) where the non-negative integers n belong to an arithmetic progression. Although almost every non-negative integer n≡ (mod t) satisfies bk(n) ≡ 0 (mod $$p_i^j$$), we show that there are infinitely many non-negative integers nr (mod t) for which bk(n) ≢ 0 (mod $$p_i^j$$) provided that there is at least one such n. Moreover the smallest such n (if there are any) is less than 2 $$\cdot 10^8 p_i^{a_i + j - 1} k^2 t^4$$.

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