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%I #37 Aug 28 2012 19:34:54
%S 2,4,8,12,34,146,706,1138,1714,1762
%N Even 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.
%C Note that F(12) = 144 = 2^4 * 3^2. - _T. D. Noe_, Aug 27 2012
%H Blair Kelly, <a href="http://mersennus.net/fibonacci">Fibonacci and Lucas factorizations</a>
%t Select[Range[2, 200, 2], 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&&i%2==0, print(i", ")))
%Y Cf. A000045, A215822, A215823, A215824.
%K nonn
%O 1,1
%A _V. Raman_, Aug 23 2012
%E Edited by _N. J. A. Sloane_, Aug 23 2012
%E 12 added by _T. D. Noe_, Aug 27 2012
%E Added 4 more terms - _V. Raman_, Aug 28 2012
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