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A080133 Conjectured number of generalized Fermat primes of the form (n+1)^2^k + n^2^k, with k>0.

%I A080133 #9 Feb 16 2025 08:32:48
%S A080133 4,2,2,2,3,2,2,2,2,1,0,3,1,3,0,1
%N A080133 Conjectured number of generalized Fermat primes of the form (n+1)^2^k + n^2^k, with k>0.
%C A080133 Primes that are the sum of consecutive integers (k=0) are excluded. Values of k <= 16 were tested. The sequence A078902 lists some of the generalized Fermat primes. Bjorn and Riesel examined generalized Fermat numbers for n <= 11 and k <= 999.
%H A080133 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 A080133 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GeneralizedFermatNumber.html">Generalized Fermat Number</a>
%e A080133 a(1) = 4 because there are four known Fermat primes (with k>0): 5, 17, 257, 65537.
%t A080133 lst={}; Do[prms=0; Do[If[PrimeQ[(n+1)^2^k+n^2^k], prms++ ], {k, 1, 16}]; AppendTo[lst, prms], {n, 16}]; lst
%Y A080133 Cf. A019434, A078902, A080131, A080134.
%K A080133 nonn,hard,more
%O A080133 1,1
%A A080133 _T. D. Noe_, Jan 30 2003