Congruences for the partition function in certain arithmetic progressions

The partition function P(n) satisfy some congruence properties. Eichhorn and Ono prove the existence of an effective constant C(m,r) (where m,r∈N have some restrictions), such that if p(mn+r)≡0(mod m) for n⩽C(m,r), then the congruence holds for every non-negative integer n. In this paper we improve the value of C(m,r) by removing its dependence in r.