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A152581
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Generalized Fermat numbers: 8^(2^n) + 1, n >= 0.
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4
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OFFSET
| 0,1
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COMMENTS
| These numbers are all composite. We re-write 8^(2^n) + 1 = (2^(2^n))^3 + 1.
Then by the identity a^n+b^n = (a+b)*(a^(n-1) - a^(n-2)*b + ... + b^(n-1)) for odd n, 2^(2^n) + 1 divides 8^(2^n) + 1. All factors of generalized Fermat numbers F_n(a,b) := a^(2^n)+b^(2^n), a >= 2, n >= 0, are of the form k*2^m+1, k >= 1, m >=0 (Riesel (1994)). - Daniel Forgues, Jun 19 2011
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LINKS
| Charles R Greathouse IV, Table of n, a(n) for n = 0..10
Anders Björn and Hans Riesel, Factors of Generalized Fermat Numbers, Mathematics of Computation, Vol. 67, No. 221, Jan., 1998, pp. 441-446.
OEIS Wiki, Generalized Fermat numbers
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FORMULA
| a(0)=9, a(n)=(a(n-1)-1)^2 + 1, n >= 1.
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EXAMPLE
| For n = 3, 8^(2^3)+1 = 16777217. Similarly, (2^8)^3 + 1 = 16777217. Then
then 2^8+1 = 257 and 16777217/257 = 65281.
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PROG
| (PARI) g(a, n) = if(a%2, b=2, b=1); for(x=0, n, y=a^(2^x)+b; print1(y", "))
(MAGMA) [8^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011
(PARI) a(n)=1<<(3*2^n)+1 \\ Charles R Greathouse IV, Jul 29 2011
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CROSSREFS
| Sequence in context: A103459 A100311 A120286 * A122733 A118465 A051375
Adjacent sequences: A152578 A152579 A152580 * A152582 A152583 A152584
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KEYWORD
| nonn,easy
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AUTHOR
| Cino Hilliard (hillcino368(AT)hotmail.com), Dec 08 2008
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EXTENSIONS
| Edited by Daniel Forgues (kephalopod(AT)gmail.com), Jun 19 2011
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