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%I #50 Sep 08 2022 08:45:01
%S 11,23,29,37,41,43,47,53,59,67,71,73,79,83,97,101,103,109,113,131,137,
%T 139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,
%U 233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331
%N Prime exponents of composite Mersenne numbers.
%C Primes p such that 2^p-1 is composite.
%C No proof is known that this sequence is infinite!
%C Assuming a conjecture of Dickson, we can prove that this sequence is infinite. See Ribenboim. - _T. D. Noe_, Jul 30 2012
%C A002515 \ {3} is a subsequence. Any proof that A002515 is infinite would imply that this sequence is infinite. - _Jeppe Stig Nielsen_, Aug 03 2020
%D Paulo Ribenboim, The New Book of Prime Number Records, Springer, 1996, p. 378.
%H Vincenzo Librandi, <a href="/A054723/b054723.txt">Table of n, a(n) for n = 1..2974</a>
%H Charles B. Barker, <a href="http://dx.doi.org/10.1090/S0002-9904-1945-08362-1">Proof that the Mersenne number M167 is composite</a>, Bull. Amer. Math. Soc. 51 (1945), 389.
%H H. S. Uhler, <a href="http://dx.doi.org/10.1090/S0002-9904-1946-08537-7">Note on the Mersenne numbers M157 and M167</a>, Bull. Amer. Math. Soc. 52 (1946), 178.
%e p=29 is included because 29 is prime, but 2^29-1 is *not* prime.
%t Select[Prime[Range[70]], ! PrimeQ[2^# - 1] &] (* _Harvey P. Dale_, Feb 03 2011 *)
%t Module[{nn=15,mp},mp=MersennePrimeExponent[Range[nn]];Complement[ Prime[ Range[ PrimePi[Last[mp]]]],mp]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Apr 10 2019 *)
%o (Magma) [p: p in PrimesUpTo(350) | not IsPrime(2^p-1)]; // _Bruno Berselli_, Oct 11 2012
%o (PARI) forprime(p=2, 1e3, if(!isprime(2^p-1), print1(p, ", "))) \\ _Felix Fröhlich_, Aug 10 2014
%Y Complement of A000043 inside A000040.
%Y Cf. A016027.
%K easy,nonn
%O 1,1
%A _Jeppe Stig Nielsen_, Apr 20 2000
%E Offset corrected by _Arkadiusz Wesolowski_, Jul 29 2012
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