A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function ${\sigma }_x(n)$

For a finite sequence of positive integers $A=\{a_j{\}}_{j=1}^k,$ we prove a recursion for divisor function $\sigma_{x}^{(A)}(n)=\sum_{d|n,\enskip d\in A}d^x.$ As a corollary, we give an affirmative solution of the problem posed in 1969 by D. B. Lahiri [3]: to find an identity for divisor function ${\sigma }_x(n)$ similar to the classic pentagonal recursion in case of $x=1.$