Problems & Puzzles: Puzzles

Puzzle 434. σ(σ(x)) = σ(x)+x Farideh Firoozbakht poses the following question: If q=2^p-1 is a Mersenne prime then q is a solution of the following equation. σ(σ(x)) = σ(x)+x  (*) Because σ(σ(q))=σ(q+1)=σ(2^p)=2^(p+1)-1=2^p+(2^p-1)=σ(q)+q. Question: Is it true that, Mersenne primes are the only solutions of (*)?

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