Do there exist a sequence $a_1, a_2, \ldots , a_n, \ldots$ of positive integers such that for any positive integers $i, j$: $$d(a_i + a_j ) = i + j?$$Here $d(n)$ is the number of positive divisors of a positive integer
Source: Belarus - Iran Competition 2022
Tags: Sequence, number theory
Do there exist a sequence $a_1, a_2, \ldots , a_n, \ldots$ of positive integers such that for any positive integers $i, j$: $$d(a_i + a_j ) = i + j?$$Here $d(n)$ is the number of positive divisors of a positive integer