%I A000215 M2503 N0990
%S A000215 3,5,17,257,65537,4294967297,18446744073709551617,340282366920938463463374607431768211457,
%T A000215 115792089237316195423570985008687907853269984665640564039457584007913129639937,
%U A000215 1340780792994259709957402499820584612747936582059239337772356144372176403007354697680187429816690342769003185\
               8186486050853753882811946569946433649006084097
%N A000215 Fermat numbers: 2^(2^n) + 1.
%C A000215 It is conjectured that just the first 5 numbers in this sequence are 
               primes.
%C A000215 An infinite coprime sequence defined by recursion. - Michael Somos Mar 
               14 2004
%C A000215 For n>0, Fermat numbers F(n) have digital roots 5 or 8 depending on whether 
               n is even or odd (Koshy). - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Mar 17 2005
%C A000215 This is the special case k=2 of sequences with exact mutual k-residues. 
               In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,
               ...,n-1}. k=1 gives Sylvester's sequence A000058. - Seppo Mustonen 
               (seppo.mustonen(AT)helsinki.fi), Sep 4 2005
%C A000215 For n>1 final two digits of a(n) are periodically repeated with period 
               4: {17, 57, 37, 97}. - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               Apr 07 2007
%C A000215 For 1<k<=2^n, a(A007814(k-1)) divides a(n)+2^k. More generally, for any 
               number k, let r=mod(k,2^n) and suppose r != 1, then a(A007814(r-1)) 
               divides a(n)+2^k. - T. D. Noe (noe(AT)sspectra.com), Jul 12 2007
%D A000215 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000215 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000215 M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 
               2nd. ed., 2001; see p. 3.
%D A000215 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 
               1976, page 7.
%D A000215 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 
               1968, vol. 2, p. 87.
%D A000215 Emmanuel Ferrand, Deformations of the Taylor Formula, Journal of Integer 
               Sequences, Vol. 10 (2007), Article 07.1.7.
%D A000215 James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
%D A000215 R. K. Guy, Unsolved Problems in Number Theory, A3.
%D A000215 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 
               3rd ed., Oxford Univ. Press, 1954, p. 14.
%D A000215 E. Hille, Analytic Function Theory, Vol. I, Chelsea, N.Y., see p. 64.
%D A000215 T. Koshy, "The Digital Root Of A Fermat Number", Journal of Recreational 
               Mathematics Vol. 32 No. 2 2002-3 Baywood NY.
%D A000215 M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag 
               NY 2001.
%D A000215 C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford 
               University Press, NY, 1966. pp. 36.
%D A000215 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 
               18, 59.
%D A000215 C. Pomerance, A tale of two sieves, Notices Amer. Math. Soc., 43 (1996), 
               1473-1485.
%D A000215 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, 
               pp. 148-9 Penguin Books 1987.
%D A000215 C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 202.
%H A000215 N. J. A. Sloane, <a href="b000215.txt">Table of n, a(n) for n = 0..11</
               a>
%H A000215 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/
               page.php/FermatNumber.html">Fermat number</a>
%H A000215 L. Euler, <a href="http://arXiv.org/abs/math.HO/0501118">Observations 
               on a theorem of Fermat and others on looking at prime numbers</a>
%H A000215 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E026.html">
               Observationes do theoremate quodam Fermatiano aliisque ad numeros 
               primos spectantibus</a>
%H A000215 Wilfrid Keller, <a href="http://www.prothsearch.net/fermat.html">Prime 
               factors k.2^n + 1 of Fermat numbers F_m</a>
%H A000215 T.-W. Leung, <a href="http://mathdb.org/resource_sharing/excalibur/Eng_v7_n4.pdf">
               A Brief Introduction to Fermat Numbers</a>
%H A000215 R. Munafo, <a href="http://www.mrob.com/pub/math/seq-a000215.html">Fermat 
               Numbers</a>
%H A000215 R. Munafo, <a href="http://www.mrob.com/pub/math/ln-notes1.html#fermat">
               Notes on Fermat numbers</a>
%H A000215 S. Mustonen, <a href="http://www.survo.fi/papers/resseq.pdf">On integer 
               sequences with mutual k-residues</a>
%H A000215 P. Sanchez, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
               FermatNumbers.html">Fermat Numbers</a>
%H A000215 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/
               Decompos/Fermat.htm">Nombres de Fermat</a>
%H A000215 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               FermatNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A000215 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GeneralizedFermatNumber.html">Generalized Fermat Number</a>
%H A000215 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fermat_number">Fermat 
               number</a>
%H A000215 Wolfram Research, <a href="http://functions.wolfram.com/04.08.03.0008.01">
               Fermat numbers are pairwise coprime</a>
%F A000215 a(0)=3, a(n) = (a(n-1)-1)^2 + 1
%F A000215 a(n) = a(n-1)*a(n-2)*...*a(1) + 2. - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Sep 15 2002
%F A000215 Conjecture : F is a Fermat prime if and only if phi(F-2) = (F-1)/2. - 
               Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 15 2002
%p A000215 A000215 := n->2^(2^n)+1;
%p A000215 with(numtheory):a[1]:=0: for n from 0 to 26 do a[n]:=fermat(n) od: seq(a[n], 
               n=0..9);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 
               21 2009]
%t A000215 lst={};Do[p=1^n+2^n;If[PrimeQ[p],AppendTo[lst,p]],{n,7!}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Jun 10 2009]
%o A000215 (PARI) a(n)=if(n<1,3*(n==0),(a(n-1)-1)^2+1)
%Y A000215 a(n) = A001146(n) + 1 = A051179(n) + 2.
%Y A000215 Cf. A019434, A050922, A051179, A063486, A073617, A085866.
%Y A000215 See A004249 for a similar sequence.
%Y A000215 Sequence in context: A067387 A050922 A070592 this_sequence A123599 A100270 
               A016045
%Y A000215 Adjacent sequences: A000212 A000213 A000214 this_sequence A000216 A000217 
               A000218
%K A000215 nonn,easy,nice
%O A000215 0,1
%A A000215 N. J. A. Sloane (njas(AT)research.att.com).