1,1

See A000043 for the values of p.

Prime repunits in base 2.

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

Mersenne primes other than the first are of form 6n+1. - Lekraj Beedassy (blekraj(AT)yahoo.com), Aug 27 2004

A034876(a(n)) = 0 and A034876(a(n)+1) = 1. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Dec 19 2004

Appears to give all n such that sigma(n+1)-sigma(n)=n - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 27 2002

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

Mersenne primes other than the first are of form 24n+7; see also A124477 - Artur Jasinski (grafix(AT)csl.pl), Nov 25 2007

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

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

Indices of triangular numbers that are also perfect numbers: A000217(a(n))=A000396(n). - Omar E. Pol (info(AT)polprimos.com), May 10 2008

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

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

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

Mersenne numbers A000225 whose indices are the prime numbers A000043. [From Omar E. Pol (info(AT)polprimos.com), Aug 31 2008]

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]

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]

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 4.

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

B. Tuckerman, The 24th Mersenne prime, Notices Amer. Math. Soc., 18 (Jun, 1971), Abstract 684-A15, p. 608.

B. Tuckerman, The 24th Mersenne prime, Proc. Nat. Acad. Sci. USA, 68 (1971), 2319-2320.

Harry J. Smith, Table of n, a(n) for n=1,...,18

P. Alfeld, The 39th Mersenne prime [From Lekraj Beedassy (blekraj(AT)yahoo.com), Nov 09 2008]

J. Bernheiden, Prime numbers(Prmality check & Mersenne primes:39-th to 43-rd)

Andrew R. Booker, The Nth Prime Page

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

D. Butler, Mersenne Primes

C. K. Caldwell, Mersenne primes

C. K. Caldwell, "Top Twenty" page, Mersenne Primes

Math Reference Project, Mersenne and Fermat Primes

L. C. Noll, Mersenne Prime Digits and Names

O. E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Primefan, The Mersenne Primes

H. J. Smith, Plot of Mersenne Primes

G. Spence, 36th Mersenne Prime Found

S. Stepney, Mersenne Prime

Thesaurus.maths.org, Mersenne Prime

B. Tuckerman, The 24th Mersenne Prime

S. S. Wagstaff, Jr., The Cunningham Project

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Perfect Number

Wikipedia, Mersenne prime

a(n) = sigma(A061652(n)) = A000203(A061652(n)). - Omar E. Pol (info(AT)polprimos.com), Apr 15 2008

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

a(n) = A000225(A000043(n)). [From Omar E. Pol (info(AT)polprimos.com), Aug 31 2008]

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

(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));

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]

Cf. A000043, A001348, A046051, A057951-A057958.

Cf. A034876.

Cf. A124477, A135659.

Cf. A019279, A061652.

Cf. A000203.

Cf. A000217.

Cf. A000225. [From Omar E. Pol (info(AT)polprimos.com), Aug 31 2008]

Adjacent sequences: A000665 A000666 A000667 this_sequence A000669 A000670 A000671

Sequence in context: A057612 A136005 A088552 this_sequence A136007 A084732 A123488

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).