OFFSET
1,1
COMMENTS
Conjecture: a(8+48*k) = 257 and a(40+48*k) = 257, where k is a nonnegative integer. - Thomas König, Feb 15 2017
Conjecture is true: 257 divides 2^(8+48*k)+1 and 2^(40+48*k)+1 but no prime < 257 ever does. Similarly, a(24+48*k) = 97. - Robert Israel, Feb 17 2017
From Robert Israel, Feb 17 2017: (Start)
If a(n) = p, there is some m such that a(n+m*j*n) = p for all j.
In particular, every member of the sequence occurs infinitely often.
a(k*n) <= a(n) for any odd k. (End)
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.
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 = 1..3967 (first 500 terms from T. D. Noe)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Thomas König, C program using gmp for testing the conjectures that a(8+k*48) = 257 and a(40+k*48) = 257
E. 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
a(n) = 3, 5, 3, 17, 3, 5, 3 for n == 1, 2, 3, 4, 5, 6, 7 (mod 8). (Proof. Let n = k*odd with k = 1, 2, or 4. As 2^k = 2, 4, 16 == -1 (mod 3, 5, 17), we get 2^n + 1 = 2^(k*odd) + 1 = (2^k)^odd + 1 == (-1)^odd + 1 == 0 (mod 3, 5, 17). Finally, 2^n + 1 !== 0 (mod p) for prime p < 3, 5, 17, respectively.) - Jonathan Sondow, Nov 28 2012
EXAMPLE
a(2^k) = 3, 5, 17, 257, 65537 is the k-th Fermat prime 2^(2^k) + 1 = A019434(k) for k = 0, 1, 2, 3, 4. - Jonathan Sondow, Nov 28 2012
MATHEMATICA
f[n_] := FactorInteger[2^n + 1][[1, 1]]; Array[f, 100] (* Robert G. Wilson v, Nov 28 2012 *)
FactorInteger[#][[1, 1]]&/@(2^Range[90]+1) (* Harvey P. Dale, Jul 25 2024 *)
PROG
(PARI) a(n) = my(m=n%8); if(m, [3, 5, 3, 17, 3, 5, 3][m], factor(2^n+1)[1, 1]); \\ Ruud H.G. van Tol, Feb 16 2024
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Jul 06 2000
Definition corrected by Jonathan Sondow, Nov 27 2012
STATUS
approved