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%I #28 Aug 28 2012 19:40:36
%S 1,2,4,7,8,12,17,23,25,34,47,49,71,73,79,89,119,137,146,151,167,191,
%T 193,199,257,271,353,359,391,409,431,433,449,569,601,706,719,751,799,
%U 809,823,833,857,881,887,929,953,1138
%N Numbers n such that the Fibonacci number F(n) can be written in the form a^2 + 3*b^2.
%C These Fibonacci numbers F(n) have no prime factor congruent to 2 mod 3 to an odd power.
%H Blair Kelly, <a href="http://mersennus.net/fibonacci">Fibonacci and Lucas factorizations</a>
%t Select[Range[200], Length[FindInstance[x^2 + 3*y^2 == Fibonacci[#], {x, y}, Integers]] > 0 &] (* _T. D. Noe_, Aug 27 2012 *)
%o (PARI) for(i=2, 500, a=factorint(fibonacci(i))~; has=0; for(j=1, #a, if(a[1, j]%3==2&&a[2, j]%2==1, has=1; break)); if(has==0, print(i", ")))
%Y Cf. A000045, A215823, A215824, A215825.
%K nonn
%O 1,2
%A _V. Raman_, Aug 23 2012
%E 1, 12, and 25 added by _T. D. Noe_, Aug 27 2012
%E Added 20 more terms - V. Raman, Aug 28 2012