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Is it known does the set of Mersenne numbers contain an infinite number of semiprimes?

By the same procedure as with primes if $n=abc$ with $a,b,c>1$ is composite number then $2^{abc}-1$ is divisible by $2^a-1$ and $2^b-1$ and $2^c-1$, so for $2^n-1$ to be a semiprime we must have $n=pq$, where $p$ and $q$ are primes and $2^p-1$ and $2^q-1$ are simulatenously primes or $n=p$, a prime.

Even if a made a mistake somewhere in description of my thoughts about the question, is this question settled anywhere?

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    $\begingroup$ AFAIK it is an open question. You might look at [OEIS sequence A085724 ](oeis.org/A085724) and references there. $\endgroup$ – Robert Israel Dec 27 '17 at 22:32

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