Non-vanishing of the partition function modulo odd primes l

Let p(n) be the usual partition function. Let l be an odd prime, and let r (mod t) be any arithmetic progression. If there exists an integer n≡r (mod t) such that p(n)≢0 (mod l), then, for large X, #{n≤X:n≡r (mod t), p(n)≢0 (mod l)} » √X/log X.