A002587 Largest prime factor of 2^n + 1.

2, 3, 5, 3, 17, 11, 13, 43, 257, 19, 41, 683, 241, 2731, 113, 331, 65537, 43691, 109, 174763, 61681, 5419, 2113, 2796203, 673, 4051, 1613, 87211, 15790321, 3033169, 1321, 715827883, 6700417, 20857, 26317, 86171, 38737, 25781083, 525313

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n) != 1 (mod n) for n = 3, 51, 141, 309, 321, 348, ... - Giovanni Resta & Thomas Ordowski, Jan 05 2014

a(n) != 1 (mod n) iff a(m) = a(n) for some m < n. Then n = 3m for m = 1, 17, 47, 103, 107, 116, ... - Thomas Ordowski, Jan 08 2014

References

• J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
• M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 2, p. 85.
• E. Lucas, Théorie des fonctions numériques simplement périodiques, Amer. J. Math., 1 (1878), 184-239, 289-321.
• 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).

Links

• Max Alekseyev, Table of n, a(n) for n = 0..1128 (terms 1..500 from T. D. Noe, terms 1037..1062 from Amiram Eldar, term 1108 from Tyler Busby)
• J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
• D. X. Charles, The abc-conjecture and the largest prime factor of 2^n + 1
• Edouard Lucas, Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240, 289-321. See pages 239 and 240.
• Edouard Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.
• S. S. Wagstaff, Jr., The Cunningham Project

Crossrefs

Cf. A000051, A002586, A006530.

Cf. similar sequences listed in A274903.

Programs

Magma

[Maximum(PrimeDivisors(2^n+1)): n in [0..40]]; // Vincenzo Librandi, Jul 12 2016

Mathematica

Table[FactorInteger[2^n + 1][[-1, 1]], {n, 0, 30}] (* Vincenzo Librandi, Jul 12 2016 *)

PARI

a(n)=my(f=factor(2^n+1)[,1]); f[#f] \\ Charles R Greathouse IV, Jul 12 2016

Formula

Charles proves that a(n) >> n^(4/3) infinitely often under the abc conjecture, and reports that Andrew Granville has improved this to a(n) >> n^2. - Charles R Greathouse IV, Apr 29 2013

a(n) = A006530(A000051(n)). - Vincenzo Librandi, Jul 12 2016

Extensions

More terms from James Sellers, Jul 06 2000

Offset 0, a(0) = 2 from Vincenzo Librandi, Jul 12 2016