Problem

Source: Iran TST 3, Day 1, Problem 3

Tags: number theory proposed, number theory



a) Does there exist a sequence $a_1<a_2<\dots$ of positive integers, such that there is a positive integer $N$ that $\forall m>N$, $a_m$ has exactly $d(m)-1$ divisors among $a_i$s? b) Does there exist a sequence $a_1<a_2<\dots$ of positive integers, such that there is a positive integer $N$ that $\forall m>N$, $a_m$ has exactly $d(m)+1$ divisors among $a_i$s?