Congruence properties for a certain kind of partition functions

In light of the modular equations of fifth and seventh order, we derive some congruence properties for a certain kind of partition functions a(n) which satisfy ∑n=0∞a(n)qn≡(q;q)∞k(modm), where k is a positive integer with 1≤k≤24 and m=2,3. In view of these properties, we obtain many infinite families of congruences for cϕk(n), the number of generalized Frobenius partitions of n with k colors, and cϕk‾(n), the number of generalized Frobenius partitions of n with k colors whose order is k under cyclic permutation of the k colors. Meanwhile, we also apply the main theorems to some other kinds of partition functions.