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? $$\sigma_k(P_a^j)+\sigma_k(P^{j+1}_a)=F(j),\tag3$$ $$j\ge1$$