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Exactly once sequence

Source: SRMO 2018 P4

August 22, 2018
number theory

Problem Statement

Does there exist a sequence of positive integers a1,a2,...a_1,a_2,... such that every positive integer occurs exactly once and that the number τ(nan+1n+(n+1)ann+1)\tau (na_{n+1}^n+(n+1)a_n^{n+1}) is divisible by nn for all positive integer.
Here τ(n)\tau (n) denotes the number of positive divisor of nn.
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