[seqfan] sigma(4n+1) mod 4 and partitions

[Georgi Guninski]

Gjergji Zaimi proved [1]:
A001935(n) mod 4 = sigma(8n+1) mod 4
and
A001936(n) mod 4 = sigma(4n+1) mod 4

A001935 Number of partitions with no even part repeated
A001936 Expansion of q^(-1/4) (eta(q^4) / eta(q))^2 in powers of q

Up to 10^7
A001935 mod 4 is zero for n = 9m+4 or 9m+7
A001936 mod 4 is zero for n = 9m+5 or 9m+8

Are A001935 or A001936 efficiently computable mod 4 - this would give sigma mod 4?

Do the recent congruences for the partitions generalize to this case?

[1] http://mathoverflow.net/questions/59178/up-to-106-sigma8n1-mod-4-oeis-a001935n-mod-4-number-of-partition/59185#59185