cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A000215 Fermat numbers: a(n) = 2^(2^n) + 1.

Original entry on oeis.org

3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, 115792089237316195423570985008687907853269984665640564039457584007913129639937
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

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, 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, Sep 04 2005
For n>1 final two digits of a(n) are periodically repeated with period 4: {17, 57, 37, 97}. - Alexander Adamchuk, 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 = k mod 2^n and suppose r != 1, then a(A007814(r-1)) divides a(n) + 2^k. - T. D. Noe, Jul 12 2007
From Daniel Forgues, Jun 20 2011: (Start)
The Fermat numbers F_n are F_n(a,b) = a^(2^n) + b^(2^n) with a = 2 and b = 1.
For n >= 2, all factors of F_n = 2^(2^n) + 1 are of the form k*(2^(n+2)) + 1 (k >= 1).
The products of distinct Fermat numbers (in their binary representation, see A080176) give rows of Sierpiński's triangle (A006943). (End)
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
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
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
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
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

Examples

			a(0) = 1*2^1 + 1 = 3 = 1*(2*1) + 1.
a(1) = 1*2^2 + 1 = 5 = 1*(2*2) + 1.
a(2) = 1*2^4 + 1 = 17 = 2*(2*4) + 1.
a(3) = 1*2^8 + 1 = 257 = 16*(2*8) + 1.
a(4) = 1*2^16 + 1 = 65537 = 2048*(2*16) + 1.
a(5) = 1*2^32 + 1 = 4294967297 = 641*6700417 = (10*(2*32) + 1)*(104694*(2*32) + 1).
a(6) = 1*2^64 + 1 = 18446744073709551617 = 274177*67280421310721 = (2142*(2*64) + 1)*(525628291490*(2*64) + 1).

References

  • 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.
  • James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, pp. 78-79.
  • 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, p. 36.
  • Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see pp. 18, 59.
  • C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 202.
  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 6-7, 70-75.
  • 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).
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 136-137.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 148-149.

Links

Crossrefs

a(n) = A001146(n) + 1 = A051179(n) + 2.
Cf. A019434, A050922, A051158, A051179, A063486, A073617, A085866.
See A004249 for a similar sequence.
Cf. A080176 for binary representation of Fermat numbers.
Cf. A220627, A220570, A220571, A125134, A249119.
Cf. A000058, A000120, A000225, A006943, A007814, A014118, A077585, A242619, A242620, A257916, A257917.
Cf. A000058, A000120, A000225, A006943, A007814, A014118, A077585, A242619, A242620, A257916, A257917.

Programs

  • Haskell
    a000215 = (+ 1) . (2 ^) . (2 ^)  -- Reinhard Zumkeller, Feb 13 2015
  • Maple
    A000215 := n->2^(2^n)+1;
  • Mathematica
    Table[2^(2^n) + 1, {n, 0, 8}] (* Alonso del Arte, Jun 07 2011 *)
  • Maxima
    A000215(n):=2^(2^n)+1$ makelist(A000215(n),n,0,10); /* Martin Ettl, Dec 10 2012 */
  • PARI
    a(n)=if(n<1,3*(n==0),(a(n-1)-1)^2+1)
  • Python
    def a(n): return 2**(2**n) + 1
print([a(n) for n in range(9)]) # Michael S. Branicky, Apr 19 2021

Formula

a(0) = 3; a(n) = (a(n-1)-1)^2 + 1, n >= 1.
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]
The above formula implies that the Fermat numbers (being all odd) are coprime.
Conjecture: F is a Fermat prime if and only if phi(F-2) = (F-1)/2. - Benoit Cloitre, Sep 15 2002
A000120(a(n)) = 2. - Reinhard Zumkeller, Aug 07 2010
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
Sum_{n>=0} 1/a(n) = A051158. - Amiram Eldar, Oct 27 2020
From Amiram Eldar, Jan 28 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = A249119.
Product_{n>=0} (1 - 1/a(n)) = 1/2. (End)
a(n) = 2*A077585(n) + 3. - César Aguilera, Jul 26 2023
a(n) = 2*2^A000225(n) + 1. - César Aguilera, Jul 11 2024

A130730 Fermat numbers of order 7 or F(n,7) = 2^(2^n)+7.

Original entry on oeis.org

9, 11, 23, 263, 65543, 4294967303, 18446744073709551623, 340282366920938463463374607431768211463, 115792089237316195423570985008687907853269984665640564039457584007913129639943
Offset: 0

Views

Author

Cino Hilliard, Jul 05 2007

Keywords

Comments

This sequence is equivalent to F(n)+ 6 or 2^(2^n)+ 1 + 6. This sequence does not appear to have any special divisibility properties. Fermat numbers of order 5 which are found in A063486, have the divisibility property if n is even, then 7 divides F(n,5). After the first 2 terms the ending digit is the same for all F(n,m) and is (6+m) mod 10.

Links

Crossrefs

Cf. A063486, A130729.

Programs

  • Magma
    [2^(2^n)+7: n in [0..11]]; // Vincenzo Librandi, Jan 09 2013
  • Mathematica
    Table[(2^(2^n) + 7), {n, 0, 15}] (* Vincenzo Librandi, Jan 09 2013 *)
  • PARI
    fplusm(n,m)= { local(x,y); for(x=0,n, y=2^(2^x)+m; print1(y",") ) }

Formula

F(n,m): The n-th Fermat number of order m = 2^(2^n)+ m. The traditional Fermat numbers are F(n,1) or Fermat numbers of order 1 in this nomenclature.

A130729 Fermat numbers of order 3 or F(n,3) = 2^(2^n)+3.

Original entry on oeis.org

5, 7, 19, 259, 65539, 4294967299, 18446744073709551619, 340282366920938463463374607431768211459, 115792089237316195423570985008687907853269984665640564039457584007913129639939
Offset: 0

Views

Author

Cino Hilliard, Jul 05 2007

Keywords

Comments

This is equivalent to F(n)+2 or 2^(2^n)+ 1 + 2. Conjecture: If n is odd, 7 is a divisor of F(n,3).
The conjecture is true: the order of 2 mod 7 is 3, and if n is odd then 2^n == 2 mod 3 so 2^(2^n) + 3 == 2^2 + 3 == 0 mod 7. - Robert Israel, Nov 20 2014

Links

Crossrefs

Cf. A063486, A130730.

Programs

  • Magma
    [2^(2^n) + 3: n in [0..11]]; // Vincenzo Librandi, Jan 09 2013
  • Mathematica
    Table[(2^(2^n) + 3), {n, 0, 15}] (* Vincenzo Librandi, Jan 09 2013 *)
  • PARI
    fplusm(n,m)= { local(x,y); for(x=0,n, y=2^(2^x)+m; print1(y",") ) }

Formula

F(n,m): The n-th Fermat number of order m = 2^(2^n)+ m. The traditional Fermat numbers are F(n,1) or Fermat numbers of order 1.

A155877 Sums of three Fermat numbers.

Original entry on oeis.org

9, 11, 13, 15, 23, 25, 27, 37, 39, 51, 263, 265, 267, 277, 279, 291, 517, 519, 531, 771, 65543, 65545, 65547, 65557, 65559, 65571, 65797, 65799, 65811, 66051, 131077, 131079, 131091, 131331, 196611, 4294967303, 4294967305, 4294967307
Offset: 1

Views

Author

Jonathan Vos Post, Jan 29 2009

Keywords

Examples

			a(1) = 3 + 3 + 3 = 9.
a(2) = 3 + 3 + 5 = 11.
a(3) = 3 + 5 + 5 = 13.
a(4) = 5 + 5 + 5 = 15.
a(5) = 3 + 3 + 17 = 23.
a(6) = 3 + 5 + 17 = 25.
a(7) = 5 + 5 + 17 = 27.
a(8) = 3 + 17 + 17 = 37.
a(9) = 5 + 17 + 17 = 39.
a(10) = 17 + 17 + 17 = 51.
a(11) = 3 + 3 + 257 = 263.

Links

Crossrefs

Cf. A000215, A019434, A050922, A051179, A059919, A063486, A073617, A078303, A078304, A085866.

Formula

{(2^(2^a) + 1) + (2^(2^b) + 1) + (2^(2^c) + 1)} = {A000215(a) + A000215(b) + A000215(c)}.

Extensions

More terms from R. J. Mathar, Feb 06 2009

A073096 Maximal element in continued fraction for s(n) = sum( k>=n,1/2^(2^k) ).

Original entry on oeis.org

6, 6, 18, 258, 65538, 4294967298, 18446744073709551618, 340282366920938463463374607431768211458, 115792089237316195423570985008687907853269984665640564039457584007913129639938
Offset: 0

Views

Author

Benoit Cloitre, Aug 18 2002

Keywords

Examples

			Continued fraction expansion of 1/2^(2^5)+1/2^(2^6)+1/2^(2^7)+...is [0, 4294967295, 4294967298, 4294967296, 4294967296,... ] where the maximum element is 4294967298, hence a(5)=4294967298

Crossrefs

Cf. A007400 for case n=0.

Formula

a(0)=6; for n>0 a(n)=2^(2^n)+2.
a(n) = A063486(n)-3 for n>0. - R. J. Mathar, Apr 22 2007
Showing 1-5 of 5 results.

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