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A078304
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Generalized Fermat numbers: 7^(2^n)+1, n >= 0.
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4
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OFFSET
| 0,1
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COMMENTS
| From Daniel Forgues, June 19 2011: (Start)
Generalized Fermat numbers F_n(a) := F_n(a,1) = a^(2^n)+1, a >= 2, n >= 0, can't be prime if a is odd (as is the case for the current sequence) (Ribenboim (1996)).
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, 1998)). (this only expresses that the factors are odd, which means that it only applies to odd generalized Fermat numbers) (End)
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..12
Eric Weisstein's World of Mathematics, Generalized Fermat Number
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) = 8, a(n)=(a(n-1)-1)^2+1, n >= 1.
a(n) = 6*a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 6*(empty product, i.e. 1)+ 2 = 8 = a(0). This means that the GCD of any pair of terms is 2. - Daniel Forgues, June 20 2011
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EXAMPLE
| a(0) = 7^1+1 = 8 = 6*(1)+2 = 6*(empty product)+2;
a(1) = 7^2+1 = 50 = 6*(8)+2;
a(2) = 7^4+1 = 2402 = 6*(8*50)+2;
a(3) = 7^8+1 = 5764802 = 6*(8*50*2402)+2;
a(4) = 7^16+1 = 33232930569602 = 6*(8*50*2402*5764802)+2;
a(5) = 7^32+1 = 1104427674243920646305299202 = 6*(8*50*2402*5764802*33232930569602)+2;
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PROG
| (MAGMA) [7^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011
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CROSSREFS
| Cf. A000215 Fermat numbers: 2^(2^n)+1, n >= 0.
Sequence in context: A124963 A195231 A162236 * A000851 A054620 A034516
Adjacent sequences: A078301 A078302 A078303 * A078305 A078306 A078307
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KEYWORD
| nonn,easy
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com), Nov 21, 2002
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EXTENSIONS
| Edited by Daniel Forgues (kephalopod(AT)gmail.com), Jun 19 2011
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