A016027 Indices of prime Mersenne numbers (A001348).

1, 2, 3, 4, 6, 7, 8, 11, 18, 24, 28, 31, 98, 111, 207, 328, 339, 455, 583, 602, 1196, 1226, 1357, 2254, 2435, 2591, 4624, 8384, 10489, 12331, 19292, 60745, 68301, 97017, 106991, 215208, 218239, 474908, 877615, 1329726, 1509263, 1622441, 1881339, 2007537, 2270720, 2584328, 2610944, 3443958

Offset: 1

Author

Robert G. Wilson v

Keywords

• nonn
• nice
• hard

Comments

The following are also terms of the sequence: 4350601, 4517402, 4811740. [updated by Amiram Eldar, Jun 11 2024].

Numbers k such that A001348(k) is a Mersenne prime A000668. - Omar E. Pol, Jul 14 2012

Numbers k such that A060286(k) is a perfect number A000396. Assuming there are no odd perfect numbers, A060286(a(n)) = A000396(n). - Omar E. Pol, Dec 13 2012

Examples

			The first four Mersenne numbers 2^2 - 1 = 3, 2^3 - 1 = 7, 2^5 - 1 = 31 and 2^7 - 1 = 127 are prime, so 1, 2, 3, 4 are members. But the fifth Mersenne number 2^11 - 1 = 2047 = 23*89 is composite, so 5 is not a member.
		

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer-Verlag, NY, 2004, Sec. A3.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford Univ. Press, 1954, p. 16.
• Paulo Ribenboim, The New Book of Prime Number Records, Springer-Verlag, NY, 1996, Chap. 2, Sec. VII.

Links

• Andrew R. Booker, The Nth Prime Page.
• Chris K. Caldwell, Mersenne Primes.
• Will Edgington, List of Mersenne primes. [from Internet Archive Wayback Machine]
• Great Internet Mersenne Prime Search (GIMPS), Distributed Computing Projects.
• Paulo Ribenboim, Galimatias arithmeticae, Mathematics Magazine, Vol. 71, No. 5 (Dec. 1998), page 337.
• Wikipedia, Mersenne prime.

Crossrefs

Cf. A000043, A000396, A001348, A059305 (index of the n-th Mersenne prime), A060286

Programs

Mathematica

a = {}; Do[If[PrimeQ[2^Prime[n] - 1], AppendTo[a, n]], {n, 1, 100}]; a (* Artur Jasinski *)
PrimePi[{(* copy the terms from A000043 *)}] (* Robert G. Wilson v, Jan 20 2014 *)
Position[Array[2^Prime[#] - 1 &, 640], ?PrimeQ][[All, 1]] (* _Michael De Vlieger, Jan 31 2018 *)
Array[PrimePi@ MersennePrimeExponent@# &, 45] (* Robert G. Wilson v, Feb 12 2018 *)

PARI

i=0; for(n=1, 1e3, if(isprime(n), i++; if(ispseudoprime(2^n-1), print1(i, ", ")))) \\ Felix Fröhlich, Aug 12 2014

Formula

a(n) = pi(A000043(n)).

a(n) = A000720(A000043(n)).

Extensions

Corrected by Vasiliy Danilov (danilovv(AT)usa.net), Jun 15 1998

Further corrections from Reto Keiser (rkeiser(AT)stud.ee.ethz.ch), Jan 10 2001

a(39) from Robert G. Wilson v, Mar 20 2006

a(40) from Robert G. Wilson v, May 29 2011

a(41) from Robert G. Wilson v, Jul 07 2012

a(42) from Robert G. Wilson v, Jan 20 2014

a(43)-a(44) from Robert G. Wilson v, Aug 20 2015

a(45) from Patrick J. McNab, Dec 18 2017

a(46)-a(47) from Ivan Panchenko, Apr 09 2018

a(48) from Amiram Eldar, Jun 11 2024