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A self-Fibonacci number is a number $n$ such that $n$ divides $Fib(n)$.

https://oeis.org/A023172

Does this sequence contain a product of any two its elements?

I didn't found this exact result among known stuff.

I did some computer analysis and found no counterexample.

I managed to prove that if $k,l$ are in the sequence and $\gcd(k,l)=1$, then $kl$ is in the sequence.

larry01
  • 1,816
  • Yes, I ask if a product of every two elements of the sequence is an element of this sequence (e.g. numbers 12 and 24 appear in the sequence, and so does $12*24=288$). – larry01 May 06 '20 at 11:36
  • Please add a proof of your last statement. – lhf May 06 '20 at 12:09
  • since $k|kl$ and $l|kl$, we have $k|Fib(k)|Fib(kl)$ and $l|Fib(l)|Fib(kl)$ (because the divisibility of indices implies the divisibility of Fibonacci numbers - this is well known), and now if $\gcd(k,l)=1$, the result follows... – larry01 May 06 '20 at 12:14

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