%I #23 Apr 03 2023 10:36:12
%S 6,26,626,390626,152587890626,23283064365386962890626,
%T 542101086242752217003726400434970855712890626
%N Generalized Fermat numbers: 5^(2^n) + 1, n >= 0.
%H Arkadiusz Wesolowski, <a href="/A199591/b199591.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 C. K. Caldwell, "Top Twenty" page, <a href="https://t5k.org/top20/page.php?id=29">Generalized Fermat Divisors (base=5)</a>.
%H Wilfrid Keller, <a href="http://www.prothsearch.com/GFN05.html">GFN05 factoring status</a>.
%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) = 6; a(n) = (a(n-1)-1)^2 + 1, n >= 1.
%F a(0) = 6, a(1) = 26; a(n) = a(n-1) + 4*5^(2^(n-1))*Product_{i=0..n-2} a(i), n >= 2.
%F a(0) = 6, a(1) = 26; a(n) = a(n-1)^2 - 2*(a(n-2)-1)^2, n >= 2.
%F a(0) = 6; a(n) = 4*(Product_{i=0..n-1} a(i)) + 2, n >= 1.
%F a(n) = A152578(n) - 1.
%F Sum_{n>=0} 2^n/a(n) = 1/4. - _Amiram Eldar_, Oct 03 2022
%e a(0) = 5^(2^0) + 1 = 5^1 + 1 = 6 = 4*(2^0) + 2;
%e a(1) = 5^(2^1) + 1 = 5^2 + 1 = 26 = 4*(2^1*3) + 2;
%e a(2) = 5^(2^2) + 1 = 5^4 + 1 = 626 = 4*(2^2*3*13) + 2;
%e a(3) = 5^(2^3) + 1 = 5^8 + 1 = 390626 = 4*(2^3*3*13*313) + 2;
%e a(4) = 5^(2^4) + 1 = 5^16 + 1 = 152587890626 = 4*(2^4*3*13*313*195313) + 2;
%e a(5) = 5^(2^5) + 1 = 5^32 + 1 = 23283064365386962890626 = 4*(2^5*3*13*313*195313*76293945313) + 2;
%t Table[5^2^n + 1, {n, 0, 6}]
%o (Magma) [5^2^n+1 : n in [0..6]]
%o (PARI) for(n=0, 6, print1(5^2^n+1, ", "))
%Y Cf. A059919, A078303, A078304, A152581, A080176, A199592, A152585.
%K easy,nonn
%O 0,1
%A _Arkadiusz Wesolowski_, Nov 08 2011