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d(n^2+1) is not monotone for n>n_0, any n_0

Source: VJIMC 2003 1.1

July 14, 2021
number theory

Problem Statement

Let d(k)d(k) denote the number of natural divisors of a natural number kk. Prove that for any natural number n0n_0 the sequence {d(n2+1)}n=n0∞\left\{d(n^2+1)\right\}^\infty_{n=n_0} is not strictly monotone.
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