A002587 Largest prime factor of 2^n + 1. (Formerly M2386 N0948)

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

0,1

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..1122 (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

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

Mathematica

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

Prog

(Magma) [Maximum(PrimeDivisors(2^n+1)): n in [0..40]]; // Vincenzo Librandi, Jul 12 2016
(PARI) a(n)=my(f=factor(2^n+1)[, 1]); f[#f] \\ Charles R Greathouse IV, Jul 12 2016

Crossrefs

Cf. A000051, A002586, A006530.

Cf. similar sequences listed in A274903.

Sequence in context: A113222 A366671 A060444 * A152814 A280319 A021981

Adjacent sequences: A002584 A002585 A002586 * A002588 A002589 A002590

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Jul 06 2000

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

Status

approved