Up to $10^6$:
$\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$
A001935 Number of partitions with no even part repeated
Is this true in general?
It would mean relation between restricted partitions of $n$ and divisors of $8n+1$.
Another one up to $10^6$ is:
$\sigma(4n+1) \mod 4 = A001936(n) \mod 4$
A001936 Expansion of q^(-1/4) (eta(q^4) / eta(q))^2 in powers of q
$\sigma(n)$ is sum of divisors of $n$.
sigma(8n+1) mod 4 starts: 1, 1, 2, 3, 0, 2, 1, 0, 0, 2, 1, 2, 2, 0, 2, 1, 0, 2, 0, 2, 0, 3, 0, 0, 2, 0, 0, 0, 3, 2
sigma(4n+1) mod 4 starts: 1, 2, 1, 2, 2, 0, 3, 2, 0, 2, 2, 2, 1, 2, 0, 2, 0, 0, 2, 0, 1, 0, 2, 0, 2, 2
Update
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
