|
%I #40 Jul 07 2023 18:56:17
%S 7,73,262657,18014398643699713,
%T 5846006549323611672814741748716771307882079584257
%N a(n) = 1 + 2^k + 4^k where k = 3^n.
%C The first three terms are prime. Are there more? Golomb shows that k must be a power of 3 in order for 1 + 2^k + 4^k to be prime. - _T. D. Noe_, Jul 16 2008
%C The next term, a(5) has 147 digits and is too large to include in DATA. - _David A. Corneth_, Aug 19 2020
%H Jeppe Stig Nielsen, <a href="/A051154/b051154.txt">Table of n, a(n) for n = 0..6</a>
%H Dario Alpern, <a href="https://www.alpertron.com.ar/MODFERM.HTM">Factors of Generalized Fermat Numbers</a>
%H Walter Feit, <a href="https://doi.org/10.1090/S0002-9939-1990-1002157-4">Finite projective planes and a question about primes</a>, Proc. AMS, Vol. 108(1990), 561-564.
%H Solomon W. Golomb, <a href="https://www.jstor.org/stable/2321679">Cyclotomic polynomials and factorization theorems</a>, Amer. Math. Monthly 85 (1978), 734-737.
%F a(n) = (2^(3^(n+1))-1)/(2^(3^n)-1).
%p F:= proc(n,r) local p; p := ithprime(r); (2^(p^(n+1))-1)/(2^(p^n)-1); end:
%p [ seq(F(n,2), n=0..5) ];
%t Table[4^(3^n) + 2^(3^n) + 1, {n, 1, 5}] (* _Artur Jasinski_, Oct 31 2011 *)
%o (PARI) a(n)=1+2^3^n+4^3^n \\ _Charles R Greathouse IV_, Oct 31 2011
%Y Cf. A001576, A051155, A051156, A051157.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
| 