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%I #42 Apr 03 2023 10:36:10
%S 11,101,10001,100000001,10000000000000001,
%T 100000000000000000000000000000001,
%U 10000000000000000000000000000000000000000000000000000000000000001
%N Generalized Fermat numbers: 10^(2^n) + 1, n >= 0.
%C As for standard Fermat numbers 2^(2^n) + 1, a number (2b)^m + 1 (with b > 1) can only be prime if m is a power of 2. On the other hand, out of the first 12 base-10 Fermat numbers, only the first two are primes.
%C Also, binary representation of Fermat numbers (in decimal, see A000215).
%H Vincenzo Librandi, <a href="/A080176/b080176.txt">Table of n, a(n) for n = 0..9</a> (shortened by _N. J. A. Sloane_, Jan 13 2019)
%H Anders Björn and Hans Riesel, <a href="http://www.jstor.org/stable/2584996">Factors of Generalized Fermat Numbers</a>, Mathematics of Computation, Vol. 67, No. 221, Jan., 1998, pp. 441-446.
%H C. K. Caldwell, "Top Twenty" page, <a href="https://t5k.org/top20/page.php?id=10">Generalized Fermat Divisors (base=10)</a>.
%H Wilfrid Keller, <a href="http://www.prothsearch.com/GFN10.html">GFN10 factoring status</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, 2012 - From N. J. A. Sloane, Jun 13 2012
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GeneralizedFermatNumber.html">Generalized Fermat Number</a>.
%H OEIS Wiki, <a href="/wiki/Generalized_Fermat_numbers">Generalized Fermat numbers</a>.
%F a(0) = 11; a(n) = (a(n - 1) - 1)^2 + 1.
%F a(n) = 9*a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 9*(empty product, i.e., 1)+ 2 = 11 = a(0). - _Daniel Forgues_, Jun 20 2011
%F Sum_{n>=0} 2^n/a(n) = 1/9. - _Amiram Eldar_, Oct 03 2022
%e a(0) = 10^1 + 1 = 11 = 9*(1) + 2 = 9*(empty product) + 2.
%e a(1) = 10^2 + 1 = 101 = 9*(11) + 2.
%e a(2) = 10^4 + 1 = 10001 = 9*(11*101) + 2.
%e a(3) = 10^8 + 1 = 100000001 = 9*(11*101*10001) + 2.
%e a(4) = 10^16 + 1 = 10000000000000001 = 9*(11*101*10001*100000001) + 2.
%e a(5) = 10^32 + 1 = 100000000000000000000000000000001 = 9*(11*101*10001*100000001*10000000000000001) + 2.
%t Table[10^2^n + 1, {n, 0, 6}] (* _Arkadiusz Wesolowski_, Nov 02 2012 *)
%o (Magma) [10^(2^n) + 1: n in [0..8]]; // _Vincenzo Librandi_, Jun 20 2011
%Y Cf. A000215 (Fermat numbers: 2^(2^n) + 1, n >= 0).
%Y Cf. A019434, A080174, A080175, A059919, A199591, A078303, A078304, A152581, A199592, A152585.
%K easy,nonn
%O 0,1
%A _Jens Voß_, Feb 04 2003
%E Edited by _Daniel Forgues_, Jun 19 2011
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