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 A078304 Generalized Fermat numbers: 7^(2^n)+1, n >= 0. 13
 8, 50, 2402, 5764802, 33232930569602, 1104427674243920646305299202, 1219760487635835700138573862562971820755615294131238402 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS From Daniel Forgues, Jun 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) LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..12 Anders Björn and Hans Riesel, Factors of Generalized Fermat Numbers, Mathematics of Computation, Vol. 67, No. 221, Jan., 1998, pp. 441-446. Eric Weisstein's World of Mathematics, Generalized Fermat Number. OEIS Wiki, Generalized Fermat numbers. 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, Jun 20 2011 Sum_{n>=0} 2^n/a(n) = 1/6. - Amiram Eldar, Oct 03 2022 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. MATHEMATICA Table[7^2^n + 1, {n, 0, 6}] (* Arkadiusz Wesolowski, Nov 02 2012 *) PROG (Magma) [7^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011 CROSSREFS Cf. A000215 (Fermat numbers: 2^(2^n)+1, n >= 0). Cf. A059919, A199591, A078303, A152581, A080176, A199592, A152585. Sequence in context: A162236 A215874 A348594 * A000851 A054620 A034516 Adjacent sequences: A078301 A078302 A078303 * A078305 A078306 A078307 KEYWORD nonn,easy AUTHOR Eric W. Weisstein, Nov 21 2002 EXTENSIONS Edited by Daniel Forgues, Jun 19 2011 STATUS approved

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