%I A000668 M2696 N1080
%S A000668 3,7,31,127,8191,131071,524287,2147483647,2305843009213693951,
%T A000668 618970019642690137449562111,162259276829213363391578010288127,
%U A000668 170141183460469231731687303715884105727
%N A000668 Mersenne primes (of form 2^p - 1 where p is a prime).
%C A000668 See A000043 for the values of p.
%C A000668 Prime repunits in base 2.
%C A000668 Define f(k) = 2k+1; begin with k = 2, a(n+1) = least prime of the form 
               f(f(f(...(a(n)))). - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 
               Dec 26 2003
%C A000668 Mersenne primes other than the first are of form 6n+1. - Lekraj Beedassy 
               (blekraj(AT)yahoo.com), Aug 27 2004
%C A000668 A034876(a(n)) = 0 and A034876(a(n)+1) = 1. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 
               Dec 19 2004
%C A000668 Appears to give all n such that sigma(n+1)-sigma(n)=n - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Aug 27 2002
%C A000668 If n is in the sequence then sigma(sigma(n))=2n+1. Is it true that this 
               sequence gives all numbers n such that sigma(sigma(n))=2n+1? - Farideh 
               Firoozbakht (mymontain(AT)yahoo.com), Aug 19 2005
%C A000668 Mersenne primes other than the first are of form 24n+7; see also A124477 
               - Artur Jasinski (grafix(AT)csl.pl), Nov 25 2007
%C A000668 It is easily proved that if n is a Mersenne prime then n+sigma(n)=sigma(sigma(n)). 
               Is it true that Mersenne primes are all the solutions of the equation 
               x+sigma(x)=sigma(sigma(x))? - Farideh Firoozbakht (mymontain(AT)yahoo.com), 
               Feb 12 2008
%C A000668 Sum of divisors of n-th even superperfect number A061652(n). Sum of divisors 
               of n-th superperfect number A019279(n), if there are no odd superperfect 
               numbers. - Omar E. Pol (info(AT)polprimos.com), Mar 11 2008
%C A000668 Indices of triangular numbers that are also perfect numbers: A000217(a(n))=A000396(n). 
               - Omar E. Pol (info(AT)polprimos.com), May 10 2008
%C A000668 Number of positive integers (1, 2, 3,...) whose sum is the n-th perfect 
               number A000396(n). - Omar E. Pol (info(AT)polprimos.com), May 10 
               2008
%C A000668 Vertex number where the n-th perfect number A000396(n) is located in 
               the square spiral whose vertices are the positive triangular numbers 
               A000217. - Omar E. Pol (info(AT)polprimos.com), May 10 2008
%C A000668 R(a(n)) is prime when R(k) means the digital reverse of k base 2. In 
               base 10, R(a(n)) is prime when R(k) means the digital reverse of 
               k base 10. For example, R(2^53-1) = 1990474529917009 is prime although 
               2^53-1 is an element of A001348 (not itself prime). - Jonathan Vos 
               Post (jvospost3(AT)gmail.com), Jul 11 2008
%C A000668 Mersenne numbers A000225 whose indices are the prime numbers A000043. 
               [From Omar E. Pol (info(AT)polprimos.com), Aug 31 2008]
%C A000668 Let p prime number. 2^p-1 is a Mersenne prime if 2^p does not belong 
               to the triangle 10; 16, 26; 22, 36, 50; 28, 46, 64, 82; 34, 56, 78, 
               100, 122; 40, 66, 92, 118, 144, 170; 46, 76, 106, 136, 166, 196, 
               226; ... [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 
               11 2009]
%C A000668 It seems that the series put up by Vincenzo Librandi fit y=ax+b such 
               that a=4x+6, and b=6x+10, I have code to figure these out in C but 
               it may need additions to deal with huge numbers.  - Roddy Macphee 
               (science_man_88(AT)yahoo.com), Nov 06 2009
%C A000668 It is possible to append a single digit to the end of any of the first 
               9 Mersenne primes, such that the resulting number is also prime. 
               [From Dmitry Kamenetsky (dkamen(AT)rsise.anu.edu.au), Jan 30 2009]
%D A000668 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 
               1976, page 4.
%D A000668 J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, 
               Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and 
               later supplements.
%D A000668 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence 
               Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A000668 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000668 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000668 B. Tuckerman, The 24th Mersenne prime, Notices Amer. Math. Soc., 18 (Jun, 
               1971), Abstract 684-A15, p. 608.
%D A000668 B. Tuckerman, The 24th Mersenne prime, Proc. Nat. Acad. Sci. USA, 68 
               (1971), 2319-2320.
%H A000668 Harry J. Smith, <a href="b000668.txt">Table of n, a(n) for n=1,...,18</
               a>
%H A000668 P. Alfeld, <a href="http://www.math.utah.edu/~pa/math/largeprime.html">
               The 39th Mersenne prime</a> [From Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Nov 09 2008]
%H A000668 J. Bernheiden, <a href="http://translate.google.com/translate?hl=en&sl=de&u=http:/
               /www.mathe-schule.de/Mathe/primzahlen.htm">Prime numbers(Prmality 
               check & Mersenne primes:39-th to 43-rd)</a>
%H A000668 Andrew R. Booker, <a href="http://primes.utm.edu/nthprime/">The Nth Prime 
               Page</a>
%H A000668 J. Brillhart et al., <a href="http://www.ams.org/online_bks/conm22/">
               Factorizations of b^n +- 1</a>, Contemporary Mathematics, Vol. 22, 
               Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
%H A000668 D. Butler, <a href="http://www.tsm-resources.com/alists/mers.html">Mersenne 
               Primes</a>
%H A000668 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/mersenne/
               index.html">Mersenne primes</a>
%H A000668 C. K. Caldwell, "Top Twenty" page, <a href="http://www.utm.edu/research/
               primes/lists/top20/Mersenne.html">Mersenne Primes</a>
%H A000668 Math Reference Project, <a href="http://www.mathreference.com/num,mers.html">
               Mersenne and Fermat Primes</a>
%H A000668 L. C. Noll, <a href="http://www.isthe.com/chongo/tech/math/prime/mersenne.html">
               Mersenne Prime Digits and Names</a>
%H A000668 O. E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica 
               de los numeros primos y perfectos</a>.
%H A000668 Primefan, <a href="http://primefan.tripod.com/MersennePrimes.html">The 
               Mersenne Primes</a>
%H A000668 H. J. Smith, <a href="http://harry-j-smith.com/hjsmithh/Perfect/MersPlot.html">
               Plot of Mersenne Primes</a>
%H A000668 G. Spence, <a href="http://www.rugeley.demon.co.uk/gimps/prime.htm">36th 
               Mersenne Prime Found</a>
%H A000668 S. Stepney, <a href="http://public.logica.com/~stepneys/cyc/m/mersenne.htm">
               Mersenne Prime</a>
%H A000668 Thesaurus.maths.org, <a href="http://thesaurus.maths.org/dictionary/map/
               word/990">Mersenne Prime</a>
%H A000668 B. Tuckerman, <a href="http://www.pnas.org/cgi/reprint/68/10/2319.pdf">
               The 24th Mersenne Prime</a>
%H A000668 S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/
               cun/index.html">The Cunningham Project</a>
%H A000668 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               MersennePrime.html">Link to a section of The World of Mathematics.</
               a>
%H A000668 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PerfectNumber.html">Perfect Number</a>
%H A000668 Wikipedia, <a href="http://www.wikipedia.org/wiki/Mersenne_prime">Mersenne 
               prime</a>
%F A000668 a(n) = sigma(A061652(n)) = A000203(A061652(n)). - Omar E. Pol (info(AT)polprimos.com), 
               Apr 15 2008
%F A000668 a(n) = sigma(A019279(n)) = A000203(A019279(n)), provided that there are 
               no odd superperfect numbers. - Omar E. Pol (info(AT)polprimos.com), 
               May 10 2008
%F A000668 a(n) = A000225(A000043(n)). [From Omar E. Pol (info(AT)polprimos.com), 
               Aug 31 2008]
%t A000668 a = {}; Do[If[DivisorSigma[1, n + 1] - DivisorSigma[1, n] == n, Print[n]; 
               AppendTo[a, n]], {n, 1, 2000000}]; a - Artur Jasinski (grafix(AT)csl.pl), 
               Dec 09 2007
%o A000668 (PARI) q(n)= { if (n==1, return(2)); if (n==2, return(3)); if (n==3, 
               return(5)); if (n==4, return(7)); if (n==5, return(13)); if (n==6, 
               return(17)); if (n==7, return(19)); if (n==8, return(31)); if (n==9, 
               return(61)); if (n==10, return(89)); if (n==11, return(107)); if 
               (n==12, return(127)); if (n==13, return(521)); if (n==14, return(607)); 
               if (n==15, return(1279)); if (n==16, return(2203)); if (n==17, return(2281)); 
               if (n==18, return(3217)); if (n==19, return(4253)); if (n==20, return(4423)); 
               if (n==21, return(9689)); if (n==22, return(9941)); if (n==23, return(11213));
%o A000668 if (n==24, return(19937)); if (n==25, return(21701)); if (n==26, return(23209)); 
               if (n==27, return(44497)); if (n==28, return(86243)); if (n==29, 
               return(110503)); if (n==30, return(132049)); if (n==31, return(216091)); 
               if (n==32, return(756839)); if (n==33, return(859433)); if (n==34, 
               return(1257787)); if (n==35, return(1398269)); if (n==36, return(2976221)); 
               if (n==37, return(3021377)); if (n==38, return(6972593)); if (n==39, 
               return(13466917)); return(0); } { for (n = 1, 18, write("b000668.txt", 
               n, " ", 2^q(n) - 1); ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               Jun 26 2009]
%Y A000668 Cf. A000043, A001348, A046051, A057951-A057958.
%Y A000668 Cf. A034876.
%Y A000668 Cf. A124477, A135659.
%Y A000668 Cf. A019279, A061652.
%Y A000668 Cf. A000203.
%Y A000668 Cf. A000217.
%Y A000668 Cf. A000225. [From Omar E. Pol (info(AT)polprimos.com), Aug 31 2008]
%Y A000668 Sequence in context: A057612 A136005 A088552 this_sequence A136007 A084732 
               A123488
%Y A000668 Adjacent sequences: A000665 A000666 A000667 this_sequence A000669 A000670 
               A000671
%K A000668 nonn,nice,new
%O A000668 1,1
%A A000668 N. J. A. Sloane (njas(AT)research.att.com).