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Let $\sigma(n)$ denote the sum of the divisors of $n$. (https://oeis.org/A000203)

It is relatively easy to find numbers $n$ such that $f(g(n)) = g(f(n))$ where $f(n) = \sigma(n)$ and $g(n) = \sigma(n) -n$ (https://oeis.org/A291881) when $n$ is even.

Question. What is the smallest odd number $n \ge 1$ such that $\sigma(\sigma(n)) = \sigma(\sigma(n)-n)+\sigma(n)$ (if such an odd number exists) ?

If someone can show that there is no such odd $n$ or there is any reference to this spesific equation, that answer is also very welcome.

Thanks.

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    $\begingroup$ The only such $n$ less than $5.7\cdot10^9$ are all even: $2, 38040, 51888, 236644, 260880, 3097024, 5283852, 5667312, 11777472, 46120848, 52981252, 69128640, 121352208, 330364848, 485906400, 662736600, 769422720, 1111869360, 1267978320, 1272335760, 1426817904, 1807128528, 2107406448, 2381691312, 2452404544, 2691587568, 3758996016, 4403660352, 5139308592.$ $\endgroup$
    – Greg Hurst
    Commented Aug 4, 2020 at 14:35
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    $\begingroup$ Thanks for your computation. It is noted that it is greater than $2.10^{11}$, if such $n$ exists. $\endgroup$
    – Alkan
    Commented Aug 4, 2020 at 16:35
  • $\begingroup$ @Alkan: What is your motivation for pursuing a solution to this problem? $\endgroup$
    – Jose Arnaldo Bebita
    Commented Sep 6, 2020 at 7:48
  • $\begingroup$ I am sorry for late response. I was curious about a possible connection to conjecture on existence of an odd perfect number and such nesting argument that question focuses on. So I wanted to share my observation on sequence entry and increase my knowledge on the subject. Best regards. $\endgroup$
    – Alkan
    Commented Jul 5, 2021 at 17:49

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