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Periodic sequence modulo f(n)

Source: 2023 Turkey TST D3 P7

March 29, 2023

functionnumber theory

Problem Statement

Let us call an integer sequence

\{ a_1,a_2, \dots \}

nice if there exist a function

f: \mathbb{Z^+} \to \mathbb{Z^+}

such that

a_i \equiv a_j \pmod{n} \iff i\equiv j \pmod{f(n)}

for all

i,j,n \in \mathbb{Z^+}

. Find all nice sequences.

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