A054723 Prime exponents of composite Mersenne numbers.

11, 23, 29, 37, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, 97, 101, 103, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331

Offset

1,1

Comments

Primes p such that 2^p-1 is composite.

No proof is known that this sequence is infinite!

Assuming a conjecture of Dickson, we can prove that this sequence is infinite. See Ribenboim. - T. D. Noe, Jul 30 2012

A002515 \ {3} is a subsequence. Any proof that A002515 is infinite would imply that this sequence is infinite. - Jeppe Stig Nielsen, Aug 03 2020

References

Paulo Ribenboim, The New Book of Prime Number Records, Springer, 1996, p. 378.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..2974

Charles B. Barker, Proof that the Mersenne number M167 is composite, Bull. Amer. Math. Soc. 51 (1945), 389.

H. S. Uhler, Note on the Mersenne numbers M157 and M167, Bull. Amer. Math. Soc. 52 (1946), 178.

Example

p=29 is included because 29 is prime, but 2^29-1 is *not* prime.

Mathematica

Select[Prime[Range[70]], ! PrimeQ[2^# - 1] &] (* Harvey P. Dale, Feb 03 2011 *)
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 *)

Prog

(Magma) [p: p in PrimesUpTo(350) | not IsPrime(2^p-1)];  // Bruno Berselli, Oct 11 2012
(PARI) forprime(p=2, 1e3, if(!isprime(2^p-1), print1(p, ", "))) \\ Felix Fröhlich, Aug 10 2014

Crossrefs

Complement of A000043 inside A000040.

Cf. A016027.

Sequence in context: A138537 A271983 A136000 * A109981 A091367 A088136

Adjacent sequences: A054720 A054721 A054722 * A054724 A054725 A054726

Keyword

easy,nonn

Author

Jeppe Stig Nielsen, Apr 20 2000

Extensions

Offset corrected by Arkadiusz Wesolowski, Jul 29 2012

Status

approved