In this paper we study restricted partition functions of the form pk(q2m+b), for certain primes q, with k = 3, 5, 7. For each k we analyze which primes q satisfy pk(q2m+b) = 0 mod k for all natural numbers m. We then prove our results by applying a common method of proof to each case.
Culek, Matthew and Knecht, Amanda, "Congruences of Restricted Partition Functions" (2002). Mathematical Sciences Technical Reports (MSTR). 62.