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%I M2503 N0990 #279 Apr 04 2024 14:28:11
%S 3,5,17,257,65537,4294967297,18446744073709551617,
%T 340282366920938463463374607431768211457,
%U 115792089237316195423570985008687907853269984665640564039457584007913129639937
%N Fermat numbers: a(n) = 2^(2^n) + 1.
%C It is conjectured that just the first 5 numbers in this sequence are primes.
%C An infinite coprime sequence defined by recursion. - _Michael Somos_, Mar 14 2004
%C For n>0, Fermat numbers F(n) have digital roots 5 or 8 depending on whether n is even or odd (Koshy). - _Lekraj Beedassy_, Mar 17 2005
%C 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_, Sep 04 2005
%C For n>1 final two digits of a(n) are periodically repeated with period 4: {17, 57, 37, 97}. - _Alexander Adamchuk_, Apr 07 2007
%C For 1 < k <= 2^n, a(A007814(k-1)) divides a(n) + 2^k. More generally, for any number k, let r = k mod 2^n and suppose r != 1, then a(A007814(r-1)) divides a(n) + 2^k. - _T. D. Noe_, Jul 12 2007
%C From _Daniel Forgues_, Jun 20 2011: (Start)
%C The Fermat numbers F_n are F_n(a,b) = a^(2^n) + b^(2^n) with a = 2 and b = 1.
%C For n >= 2, all factors of F_n = 2^(2^n) + 1 are of the form k*(2^(n+2)) + 1 (k >= 1).
%C The products of distinct Fermat numbers (in their binary representation, see A080176) give rows of Sierpiński's triangle (A006943). (End)
%C Let F(n) be a Fermat number. For n > 2, F(n) is prime if and only if 5^((F(n)-1)/4) == sqrt(F(n)-1) (mod F(n)). - _Arkadiusz Wesolowski_, Jul 16 2011
%C Conjecture: let the smallest prime factor of Fermat number F(n) be P(F(n)). If F(n) is composite, then P(F(n)) < 3*2^(2^n/2 - n - 2). - _Arkadiusz Wesolowski_, Aug 10 2012
%C The Fermat primes are not Brazilian numbers, so they belong to A220627, but the Fermat composites are Brazilian numbers so they belong to A220571. For a proof, see Proposition 3 page 36 on "Les nombres brésiliens" in Links. - _Bernard Schott_, Dec 29 2012
%C It appears that this sequence is generated by starting with a(0)=3 and following the rule "Write in binary and read in base 4". For an example of "Write in binary and read in ternary", see A014118. - _John W. Layman_, Jul 30 2013
%C Conjecture: the numbers > 5 in this sequence, i.e., 2^2^k + 1 for k>1, are exactly the numbers n such that (n-1)^4-1 divides 2^(n-1)-1. - _M. F. Hasler_, Jul 24 2015
%D M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 2nd. ed., 2001; see p. 3.
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 7.
%D P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 87.
%D James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
%D R. K. Guy, Unsolved Problems in Number Theory, A3.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 14.
%D E. Hille, Analytic Function Theory, Vol. I, Chelsea, N.Y., see p. 64.
%D T. Koshy, "The Digital Root Of A Fermat Number", Journal of Recreational Mathematics Vol. 32 No. 2 2002-3 Baywood NY.
%D M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag NY 2001.
%D C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966, p. 36.
%D Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see pp. 18, 59.
%D C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 202.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 148-149.
%H N. J. A. Sloane, <a href="/A000215/b000215.txt">Table of n, a(n) for n = 0..11</a>
%H Richard Bellman, <a href="http://dx.doi.org/10.1090/S0002-9904-1947-08879-0">A note on relatively prime sequences</a>, Bull. Amer. Math. Soc. 53 (1947), 778-779.
%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php/FermatNumber.html">Fermat number</a>.
%H C. K. Caldwell, The prime pages <a href="https://primes.utm.edu/notes/proofs/SquareMerDiv.html">All prime-square Mersenne divisors are Wieferich</a> (2021).
%H Shane Chern, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Chern/chern2.html">Fermat Numbers in Multinomial Coefficients</a>, J. Int. Seq. 17 (2014) # 14.3.2.
%H Leonhard Euler, <a href="https://arxiv.org/abs/math/0501118">Observations on a theorem of Fermat and others on looking at prime numbers</a>, arXiv:math/0501118 [math.HO], 2005-2008.
%H Leonhard Euler, <a href="http://math.dartmouth.edu/~euler/pages/E026.html">Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus</a>.
%H Emmanuel Ferrand, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Ferrand/ferrand8.html">Deformations of the Taylor Formula</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.7.
%H Richard K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%H Wilfrid Keller, <a href="http://www.prothsearch.com/fermat.html">Prime factors k.2^n + 1 of Fermat numbers F_m</a>
%H Jiří Klaška, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Klaska/klaska6.html">Jakóbczyk's Hypothesis on Mersenne Numbers and Generalizations of Skula's Theorem</a>, J. Int. Seq., Vol. 26 (2023), Article 23.3.8.
%H T.-W. Leung, <a href="http://mathdb.org/resource_sharing/excalibur/Eng_v7_n4.pdf">A Brief Introduction to Fermat Numbers</a>
%H Romeo Meštrović, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - From _N. J. A. Sloane_, Jun 13 2012
%H Romeo Meštrović, <a href="https://www.researchgate.net/publication/329844912_GOLDBACH-TYPE_CONJECTURES_ARISING_FROM_SOME_ARITHMETIC_PROGRESSIONS">Goldbach-type conjectures arising from some arithmetic progressions</a>, University of Montenegro, 2018.
%H Romeo Meštrović, <a href="https://arxiv.org/abs/1901.07882">Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes</a>, arXiv:1901.07882 [math.NT], 2019.
%H Michael A. Morrison and John Brillhart, <a href="http://dx.doi.org/10.1090/S0002-9904-1971-12711-8">The factorization of F_7</a>, Bull. Amer. Math. Soc. 77 (1971), 264.
%H Robert Munafo, <a href="http://www.mrob.com/pub/seq/a000215.html">Fermat Numbers</a>
%H Robert Munafo, <a href="http://www.mrob.com/pub/math/ln-notes1.html#fermat">Notes on Fermat numbers</a>
%H Seppo Mustonen, <a href="http://www.survo.fi/papers/resseq.pdf">On integer sequences with mutual k-residues</a>.
%H Seppo Mustonen, <a href="/A000215/a000215.pdf">On integer sequences with mutual k-residues</a>. [Local copy]
%H OEIS Wiki, <a href="/wiki/Fermat_numbers">Fermat numbers</a>.
%H OEIS Wiki, <a href="/wiki/Sierpinski's_triangle">Sierpinski's triangle</a>.
%H G. A. Paxson, <a href="http://dx.doi.org/10.1090/S0025-5718-1961-0124264-0">The compositeness of the thirteenth Fermat number</a>, Math. Comp. 15 (76) (1961) 420-420.
%H Carl Pomerance, <a href="http://www.ams.org/notices/199612/pomerance.pdf">A tale of two sieves</a>, Notices Amer. Math. Soc., 43 (1996), 1473-1485.
%H P. Sanchez, PlanetMath.org, <a href="https://planetmath.org/fermatnumbers">Fermat Numbers</a>
%H Bernard Schott, <a href="http://quadrature.info/">Les nombres brésiliens</a>, Quadrature, no. 76, avril-juin 2010, pages 30-38. <a href="/A125134/a125134.pdf">Local copy</a>, included here with permission from the editors of Quadrature.
%H G. Villemin's Almanach of Numbers, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Decompos/Fermat.htm">Nombres de Fermat</a>.
%H Le Roy J. Warren, Henry G. Bray, <a href="https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-22/issue-3/On-the-square-freeness-of-Fermat-and-Mersenne-numbers/pjm/1102992105.pdf"> On the square-freeness of Fermat and Mersenne Numbers</a>, Pac. J. Math. 22 (3) (1967) 563.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatNumber.html">Fermat Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GeneralizedFermatNumber.html">Generalized Fermat Number</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fermat_number">Fermat number</a>.
%H Wolfram Research, <a href="http://functions.wolfram.com/04.08.03.0008.01">Fermat numbers are pairwise coprime</a>.
%F a(0) = 3; a(n) = (a(n-1)-1)^2 + 1, n >= 1.
%F a(n) = a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get the empty product, i.e., 1, plus 2, giving 3 = a(0). - _Benoit Cloitre_, Sep 15 2002 [edited by _Daniel Forgues_, Jun 20 2011]
%F The above formula implies that the Fermat numbers (being all odd) are coprime.
%F Conjecture: F is a Fermat prime if and only if phi(F-2) = (F-1)/2. - _Benoit Cloitre_, Sep 15 2002
%F A000120(a(n)) = 2. - _Reinhard Zumkeller_, Aug 07 2010
%F If a(n) is composite, then a(n) = A242619(n)^2 + A242620(n)^2 = A257916(n)^2 - A257917(n)^2. - _Arkadiusz Wesolowski_, May 13 2015
%F Sum_{n>=0} 1/a(n) = A051158. - _Amiram Eldar_, Oct 27 2020
%F From _Amiram Eldar_, Jan 28 2021: (Start)
%F Product_{n>=0} (1 + 1/a(n)) = A249119.
%F Product_{n>=0} (1 - 1/a(n)) = 1/2. (End)
%F a(n) = 2*A077585(n) + 3. - _César Aguilera_, Jul 26 2023
%e a(0) = 1*2^1 + 1 = 3 = 1*(2*1) + 1.
%e a(1) = 1*2^2 + 1 = 5 = 1*(2*2) + 1.
%e a(2) = 1*2^4 + 1 = 17 = 2*(2*4) + 1.
%e a(3) = 1*2^8 + 1 = 257 = 16*(2*8) + 1.
%e a(4) = 1*2^16 + 1 = 65537 = 2048*(2*16) + 1.
%e a(5) = 1*2^32 + 1 = 4294967297 = 641*6700417 = (10*(2*32) + 1)*(104694*(2*32) + 1).
%e a(6) = 1*2^64 + 1 = 18446744073709551617 = 274177*67280421310721 = (2142*(2*64) + 1)*(525628291490*(2*64) + 1).
%p A000215 := n->2^(2^n)+1;
%t Table[2^(2^n) + 1, {n, 0, 8}] (* _Alonso del Arte_, Jun 07 2011 *)
%o (PARI) a(n)=if(n<1,3*(n==0),(a(n-1)-1)^2+1)
%o (Maxima) A000215(n):=2^(2^n)+1$ makelist(A000215(n),n,0,10); /* _Martin Ettl_, Dec 10 2012 */
%o (Haskell)
%o a000215 = (+ 1) . (2 ^) . (2 ^) -- _Reinhard Zumkeller_, Feb 13 2015
%o (Python)
%o def a(n): return 2**(2**n) + 1
%o print([a(n) for n in range(9)]) # _Michael S. Branicky_, Apr 19 2021
%Y a(n) = A001146(n) + 1 = A051179(n) + 2.
%Y Cf. A019434, A050922, A051158, A051179, A063486, A073617, A085866.
%Y See A004249 for a similar sequence.
%Y Cf. A080176 for binary representation of Fermat numbers.
%Y Cf. A220627, A220570, A220571, A125134, A249119.
%K nonn,easy,nice
%O 0,1
%A _N. J. A. Sloane_
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