Take care of mod and sum

If $\sum_{n=1}^{\infty} \frac{(n\bmod k)}{n^2+n}=\ln(pk), \; p, k\in\mathbb N$ and $\sum_{k=2}^{\infty}\sum_{n=1}^{\infty} \frac{n\bmod(2k)}{k^3(n^2+n)}=\ln(a)\left(\zeta(b)-c\right)-\zeta^{\prime}(b)$ where $a,b,c$ are positive integers then find the value of $(a+b+c+p)^3$ .

Notation: Here $\zeta(.)$ is zeta function and $\zeta^{\prime}(.)$ is derivatives of zeta function.

This is an original problem and generalization of Take care of just 1 and 2 .