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%I #121 Jun 11 2024 04:43:36
%S 1,2,3,4,6,7,8,11,18,24,28,31,98,111,207,328,339,455,583,602,1196,
%T 1226,1357,2254,2435,2591,4624,8384,10489,12331,19292,60745,68301,
%U 97017,106991,215208,218239,474908,877615,1329726,1509263,1622441,1881339,2007537,2270720,2584328,2610944,3443958
%N Indices of prime Mersenne numbers (A001348).
%C The following are also terms of the sequence: 4350601, 4517402, 4811740. [updated by _Amiram Eldar_, Jun 11 2024].
%C Numbers k such that A001348(k) is a Mersenne prime A000668. - _Omar E. Pol_, Jul 14 2012
%C 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
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer-Verlag, NY, 2004, Sec. A3.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford Univ. Press, 1954, p. 16.
%D Paulo Ribenboim, The New Book of Prime Number Records, Springer-Verlag, NY, 1996, Chap. 2, Sec. VII.
%H Andrew R. Booker, <a href="https://t5k.org/nthprime/">The Nth Prime Page</a>.
%H Chris K. Caldwell, <a href="https://primes.utm.edu/mersenne/index.html?id=research&month=primes&day=mersenne&year=index">Mersenne Primes</a>.
%H Will Edgington, <a href="https://web.archive.org/web/20060630100606/http://www.garlic.com:80/~wedgingt/primeM.txt">List of Mersenne primes</a>. [from Internet Archive Wayback Machine]
%H Great Internet Mersenne Prime Search (GIMPS), <a href="http://www.mersenne.org/">Distributed Computing Projects</a>.
%H Paulo Ribenboim, <a href="http://www.jstor.org/stable/2690773">Galimatias arithmeticae</a>, Mathematics Magazine, Vol. 71, No. 5 (Dec. 1998), page 337.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Mersenne_prime"> Mersenne prime</a>.
%F a(n) = pi(A000043(n)).
%F a(n) = A000720(A000043(n)).
%e 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.
%t a = {}; Do[If[PrimeQ[2^Prime[n] - 1], AppendTo[a, n]], {n, 1, 100}]; a (* _Artur Jasinski_ *)
%t PrimePi[{(* copy the terms from A000043 *)}] (* _Robert G. Wilson v_, Jan 20 2014 *)
%t Position[Array[2^Prime[#] - 1 &, 640], _?PrimeQ][[All, 1]] (* _Michael De Vlieger_, Jan 31 2018 *)
%t Array[PrimePi@ MersennePrimeExponent@# &, 45] (* _Robert G. Wilson v_, Feb 12 2018 *)
%o (PARI) i=0; for(n=1, 1e3, if(isprime(n), i++; if(ispseudoprime(2^n-1), print1(i, ", ")))) \\ _Felix Fröhlich_, Aug 12 2014
%Y Cf. A000043, A000396, A001348, A059305 (index of the n-th Mersenne prime), A060286
%K nonn,nice,hard
%O 1,2
%A _Robert G. Wilson v_
%E Corrected by Vasiliy Danilov (danilovv(AT)usa.net), Jun 15 1998
%E Further corrections from Reto Keiser (rkeiser(AT)stud.ee.ethz.ch), Jan 10 2001
%E a(39) from _Robert G. Wilson v_, Mar 20 2006
%E a(40) from _Robert G. Wilson v_, May 29 2011
%E a(41) from _Robert G. Wilson v_, Jul 07 2012
%E a(42) from _Robert G. Wilson v_, Jan 20 2014
%E a(43)-a(44) from _Robert G. Wilson v_, Aug 20 2015
%E a(45) from _Patrick J. McNab_, Dec 18 2017
%E a(46)-a(47) from _Ivan Panchenko_, Apr 09 2018
%E a(48) from _Amiram Eldar_, Jun 11 2024