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 $\sigma_x(n)$

TL;DR

This paper develops a recursion for a divisor function restricted to a finite sequence of integers and solves Lahiri's 1969 problem by deriving a pentagonal-like identity for the classical divisor function.

Contribution

It introduces a new recursion for the divisor function over specific sequences and provides an affirmative solution to Lahiri's longstanding problem.

Findings

Derived a recursion for $\sigma_x^{(A)}(n)$ over finite sequences.

Provided a pentagonal-like identity for the classical divisor function $\sigma_x(n)$.

Solved Lahiri's problem from 1969 with a new divisor function identity.

Abstract

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 $x=1.$