Generalise of $\sigma_k(P_a^j)+\sigma_k(P^{j+1}_a)?$

Where $ P_a$ is a prime number and $ \sigma_k(n)$ is divisor function

We encountered these two identities:

$ $ \sigma_k(P_a)+\sigma_k(P^2_a)=(P_a^k+1)^2+1\tag1$ $

$ $ \sigma_k(P_a^2)+\sigma_k(P^3_a)=(P_a^k+1)^3-(P_a^k+1)^2+(P_a^k+1)+1\tag2$ $

We would like to know if this identity can be generalise? $ $ \sigma_k(P_a^j)+\sigma_k(P^{j+1}_a)=F(j),\tag3$ $ $ j\ge1$