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

Author

Jeppe Stig Nielsen, Apr 20 2000

Keywords

• easy
• nonn

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

Examples

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

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.

Crossrefs

Complement of A000043 inside A000040.

Cf. A016027.

Programs

Magma

[p: p in PrimesUpTo(350) | not IsPrime(2^p-1)];  // Bruno Berselli, Oct 11 2012

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 *)

PARI

forprime(p=2, 1e3, if(!isprime(2^p-1), print1(p, ", "))) \\ Felix Fröhlich, Aug 10 2014

Extensions

Offset corrected by Arkadiusz Wesolowski, Jul 29 2012