Search: id:A000668 Results 1-1 of 1 results found. %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, Table of n, a(n) for n=1,...,18 %H A000668 P. Alfeld, The 39th Mersenne prime [From Lekraj Beedassy (blekraj(AT)yahoo.com), Nov 09 2008] %H A000668 J. Bernheiden, Prime numbers(Prmality check & Mersenne primes:39-th to 43-rd) %H A000668 Andrew R. Booker, The Nth Prime Page %H A000668 J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002. %H A000668 D. Butler, Mersenne Primes %H A000668 C. K. Caldwell, Mersenne primes %H A000668 C. K. Caldwell, "Top Twenty" page, Mersenne Primes %H A000668 Math Reference Project, Mersenne and Fermat Primes %H A000668 L. C. Noll, Mersenne Prime Digits and Names %H A000668 O. E. Pol, Determinacion geometrica de los numeros primos y perfectos. %H A000668 Primefan, The Mersenne Primes %H A000668 H. J. Smith, Plot of Mersenne Primes %H A000668 G. Spence, 36th Mersenne Prime Found %H A000668 S. Stepney, Mersenne Prime %H A000668 Thesaurus.maths.org, Mersenne Prime %H A000668 B. Tuckerman, The 24th Mersenne Prime %H A000668 S. S. Wagstaff, Jr., The Cunningham Project %H A000668 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000668 Eric Weisstein's World of Mathematics, Perfect Number %H A000668 Wikipedia, Mersenne prime %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). Search completed in 0.004 seconds