# Is there a positive odd $n$ such that $\sigma(\sigma(n)) = \sigma(\sigma(n)-n)+\sigma(n)$?

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.

• The only such $n$ less than $5.7\cdot10^9$ are all even: $2, 38040, 51888, 236644, 260880, 3097024, 5283852, 5667312, 11777472, 46120848, 52981252, 69128640, 121352208, 330364848, 485906400, 662736600, 769422720, 1111869360, 1267978320, 1272335760, 1426817904, 1807128528, 2107406448, 2381691312, 2452404544, 2691587568, 3758996016, 4403660352, 5139308592.$ – Chip Hurst Aug 4 at 14:35
• Thanks for your computation. It is noted that it is greater than $2.10^{11}$, if such $n$ exists. – Alkan Aug 4 at 16:35
• @Alkan: What is your motivation for pursuing a solution to this problem? – Arnie Bebita-Dris Sep 6 at 7:48