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Let $a_n$ be a sequence such that $a_1=1$ and for each $n \geq 1$ $a_{n+1}$ is the smallest positive integer distinct from $a_1,a_2,...,a_n$ such that $\gcd(a_{n+1}a_n+1,a_i)=1$ for each $i=1,2,...,n$. How to prove that every positive integer appears in $a_n$?

This question was asked on a Brazilian constest. According to one of the contestants the official solution was flawed and the problem remains open. Is this result true?

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    $\begingroup$ "According to one of the contestants the official solution was flawed and the problem remains open." Very cool! $\endgroup$
    – mathworker21
    Commented Oct 26 at 14:12
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    $\begingroup$ First few terms (not on OEIS): 1, 2, 3, 4, 6, 5, 8, 9, 10, 7, 16, 12, 13, 24, 14, 15, 18, 11, 30, 19, 22, 21, 20, 23, 26, 17, 36, 28, 25, 42, 31, 46, 43, 40, 33, 34, 27, 44, 29, 32, 38, 39, 48, 35, 56, 41, 62, 45, 50, 47... $\endgroup$
    – YCor
    Commented Oct 26 at 15:02
  • $\begingroup$ It will be added soon as oeis.org/A377362 $\endgroup$
    – Max Alekseyev
    Commented Oct 26 at 18:18
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    $\begingroup$ Maybe one can define a "height" function $h$ defined by $h(\prod_{i}p_{i}^{a_{i}}):=\sum_{i}i.a_{i}$ and see if the terms of the sequence are arranged in increasing order of their heights. $\endgroup$
    – Sylvain JULIEN
    Commented Oct 29 at 21:35

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