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Coprime to sum of terms before it

Source: China second round (A) 2018 Q4

May 5, 2019

number theorySequence

Problem Statement

Define sequence

\{a_n\}

a_1

is any positive integer, and for any positive integer

n\ge 1

a_{n+1}

is the smallest positive integer coprime to

\sum_{i=1}^{n} a_i

and not equal to

a_1,\ldots, a_n

. Prove that every positive integer appears in the sequence

\{a_n\}