Distribution of powers of the partition function modulo ℓ^j

In this paper we study Newman's conjecture for powers of the partition function. While this conjecture is known for powers of primes ℓ that are not exceptional for the power under consideration, it is an open problem for exceptional primes. We settle this conjecture in many cases for small powers of the partition function by generalizing results of Ono and Ahlgren. It should be noted our method requires a case by case examination of each power and does not yield a general method for dealing with different powers simultaneously.