A199592 - OEIS

%I #17 Oct 03 2022 04:19:32

%S 12,122,14642,214358882,45949729863572162,

%T 2111377674535255285545615254209922,

%U 4457915684525902395869512133369841539490161434991526715513934826242

%N Generalized Fermat numbers: 11^(2^n) + 1, n >= 0.

%H Arkadiusz Wesolowski, <a href="/A199592/b199592.txt">Table of n, a(n) for n = 0..11</a>

%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 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) = 12; a(n) = (a(n-1)-1)^2 + 1, n >= 1.

%F a(0) = 12, a(1) = 122; a(n) = a(n-1) + 10*11^(2^(n-1))*Product_{i=0..n-2} a(i), n >= 2.

%F a(0) = 12, a(1) = 122; a(n) = a(n-1)^2 - 2*(a(n-2)-1)^2, n >= 2.

%F a(0) = 12; a(n) = 10*(Product_{i=0..n-1} a(i)) + 2, n >= 1.

%F a(n) = A152583(n) - 1.

%F Sum_{n>=0} 2^n/a(n) = 1/10. - _Amiram Eldar_, Oct 03 2022

%e a(0) = 11^(2^0) + 1 = 11^1 + 1 = 12 = 10*(2^0) + 2;

%e a(1) = 11^(2^1) + 1 = 11^2 + 1 = 122 = 10*(2^1*6) + 2;

%e a(2) = 11^(2^2) + 1 = 11^4 + 1 = 14642 = 10*(2^2*6*61) + 2;

%e a(3) = 11^(2^3) + 1 = 11^8 + 1 = 214358882 = 10*(2^3*6*61*7321) + 2;

%e a(4) = 11^(2^4) + 1 = 11^16 + 1 = 45949729863572162 = 10*(2^4*6*61*7321*107179441) + 2;

%e a(5) = 11^(2^5) + 1 = 11^32 + 1 = 2111377674535255285545615254209922 = 10*(2^5*6*61*7321*107179441*22974864931786081) + 2;

%t Table[11^2^n + 1, {n, 0, 6}]

%o (Magma) [11^2^n+1 : n in [0..6]]

%o (PARI) for(n=0, 6, print1(11^2^n+1, ", "))

%Y Cf. A059919, A199591, A078303, A078304, A152581, A080176, A152585.

%K easy,nonn

%O 0,1

%A _Arkadiusz Wesolowski_, Nov 08 2011