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For a finite sequence of positive integers $A=\{a_j\}_{j=1}^{k},$ we prove arecursion for divisor function $\sigma_{x}^{(A)}(n)=\sum_{d|n,\enskip d\inA}d^x.$ As a corollary, we give an affirmative solution of the problem posed in1969 by D. B. Lahiri [3]: to find an identity for divisor function$\sigma_x(n)$ similar to the classic pentagonal recursion in case of $x=1.$

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 $\sigma_x(n)$