0,1

It is conjectured that just the first 5 numbers in this sequence are primes.

An infinite coprime sequence defined by recursion. - Michael Somos Mar 14 2004

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

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

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

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

M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 2nd. ed., 2001; see p. 3.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 7.

P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 87.

Emmanuel Ferrand, Deformations of the Taylor Formula, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.7.

James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).

R. K. Guy, Unsolved Problems in Number Theory, A3.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 14.

E. Hille, Analytic Function Theory, Vol. I, Chelsea, N.Y., see p. 64.

T. Koshy, "The Digital Root Of A Fermat Number", Journal of Recreational Mathematics Vol. 32 No. 2 2002-3 Baywood NY.

M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag NY 2001.

C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966. pp. 36.

C. Pomerance, A tale of two sieves, Notices Amer. Math. Soc., 43 (1996), 1473-1485.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 148-9 Penguin Books 1987.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 18, 59.

N. J. A. Sloane, Table of n, a(n) for n = 0..13

C. K. Caldwell, The Prime Glossary, Fermat number

L. Euler, Observations on a theorem of Fermat and others on looking at prime numbers

L. Euler, Observationes do theoremate quodam Fermatiano aliisque ad numeros primos spectantibus

Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m

T.-W. Leung, A Brief Introduction to Fermat Numbers

R. Munafo, Fermat Numbers

R. Munafo, Notes on Fermat numbers

S. Mustonen, On integer sequences with mutual k-residues

P. Sanchez, PlanetMath.org, Fermat Numbers

G. Villemin's Almanach of Numbers, Nombres de Fermat

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

Eric Weisstein's World of Mathematics, Generalized Fermat Number

Wikipedia, Fermat number

Wolfram Research, Fermat numbers are pairwise coprime

a(0)=3, a(n) = (a(n-1)-1)^2 + 1

a(n) = a(n-1)*a(n-2)*...*a(1) + 2. - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 15 2002

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

A000215 := n->2^(2^n)+1;

(PARI) a(n)=if(n<1, 3*(n==0), (a(n-1)-1)^2+1)

a(n) = A001146(n) + 1 = A051179(n) + 2.

Cf. A019434, A050922, A051179, A063486, A073617, A085866.

See A004249 for a similar sequence.

Adjacent sequences: A000212 A000213 A000214 this_sequence A000216 A000217 A000218

Sequence in context: A050922 A067387 A070592 this_sequence A123599 A100270 A016045

nonn,easy,nice

njas