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%I #32 Aug 11 2014 22:45:49
%S 1,2,3,4,9,11,17,19,20,22,26,27,28,33,41,43,46,51,52,57,67,68,73,76,
%T 81,83,99,113,116,118,121,123,129,139,140,153,164,171,172,194,201,219,
%U 241,242,243,244,249,281,283,284,297,313,314,316,323,339,353,356,358,362,363,369,379,382,387,401,404,417,428
%N Numbers n such that the Lucas number L(n) can be written in the form a^2 + 3*b^2.
%C These Lucas numbers L(n) have no prime factor congruent to 2 mod 3 to an odd power.
%H V. Raman, <a href="/A215810/b215810.txt">Table of n, a(n) for n = 1..109</a>
%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 == LucasL[#], {x, y}, Integers]] > 0 &] (* _T. D. Noe_, Aug 27 2012 *)
%o (PARI) for(i=2, 500, a=factorint(fibonacci(i-1)+fibonacci(i+1))~; 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. A000032, A215811, A215815, A215816.
%K nonn
%O 1,2
%A _V. Raman_, Aug 23 2012
%E Corrected by _T. D. Noe_, Aug 27 2012
%E Added 18 more terms - _V. Raman_, Aug 28 2012