irregularities in partition function modulo n

Question

It is an open problem whether the partition function is even half the time. Inspired by this, I wrote some Sage/Python code to check how many times $p(n)$ hits each residue class:

def partitionmod(n, k):
    bins = [0,] * k
    for i in range(n):
        bins[mod(Partitions(i+1).cardinality(), k)] += 1
    binstotal = 0
    for x in bins:
        binstotal += x
    for x in bins:
        print(N(x/binstotal, 30))

For $n = 50000$ and $k = 5$, I found something odd: the residue class of $0$ is hit with likelihood $.364$, which is double the odds of hitting any other residue class mod 5. Something similar happens with $$k = 7, 10, 11, 14, 15, 20, 21, 22, 25, 28, 30,\dots$$ Is there something special about multiples of 5, 7, and 11 that I'm missing?