A215816 Odd numbers n such that the Lucas number L(n) can be written in the form a^2 + 3*b^2.

1, 3, 9, 11, 17, 19, 27, 33, 41, 43, 51, 57, 67, 73, 81, 83, 99, 113, 121, 123, 129, 139, 153, 171, 201, 219, 241, 243, 249, 281, 283, 297, 313, 323, 339, 353, 363, 369, 379, 387, 401, 417, 443, 457, 459, 473, 513, 569, 571, 577, 603, 617, 657, 723, 729, 747, 843, 849, 857, 881, 891, 939, 969, 1009

Offset

1,2

Comments

These Lucas numbers L(n) have no prime factor congruent to 2 (mod 3) to an odd power.

Links

V. Raman, Table of n, a(n) for n = 1..70

Blair Kelly, Fibonacci and Lucas factorizations

Mathematica

Select[Range[1, 200, 2], Length[FindInstance[x^2 + 3*y^2 == LucasL[#], {x, y}, Integers]] > 0 &] (* T. D. Noe, Aug 27 2012 *)

Prog

(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==1, print(i", ")))

Crossrefs

Cf. A000032, A215810, A215811, A215815.

Sequence in context: A365082 A201544 A225771 * A211864 A157139 A050538

Adjacent sequences: A215813 A215814 A215815 * A215817 A215818 A215819

Keyword

nonn

Author

V. Raman, Aug 23 2012

Extensions

Corrected by T. D. Noe, Aug 27 2012

Added 32 more terms - V. Raman, Aug 28 2012

Status

approved