A215815 - OEIS

A215815

Even numbers n such that the Lucas number L(n) can be written in the form a^2 + 3*b^2.

%I #34 Apr 15 2017 08:17:11

%S 2,4,20,22,26,28,46,52,68,76,116,118,140,164,172,194,242,244,284,314,

%T 316,356,358,362,382,404,428,458,478,598,698,746,772,794,812,914,988,

%U 1004,1082

%N Even 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 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 == LucasL[#], {x, y}, Integers]] > 0 &] (* _G. C. Greubel_, Apr 14 2017 *)

%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&&i%2==0, print(i", ")))

%Y Cf. A000032, A215810, A215811, A215816.

%K nonn,more

%O 1,1

%A _V. Raman_, Aug 23 2012

%E a(22)-a(39) from _V. Raman_, Aug 28 2012