A215822 Numbers k such that the Fibonacci number F(k) can be written in the form a^2 + 3*b^2.

1, 2, 4, 7, 8, 12, 17, 23, 25, 34, 47, 49, 71, 73, 79, 89, 119, 137, 146, 151, 167, 191, 193, 199, 257, 271, 353, 359, 391, 409, 431, 433, 449, 569, 601, 706, 719, 751, 799, 809, 823, 833, 857, 881, 887, 929, 953, 1138

Offset

1,2

Comments

These Fibonacci numbers F(k) have no prime factor congruent to 2 mod 3 to an odd power.

Links

Table of n, a(n) for n=1..48.

Blair Kelly, Fibonacci and Lucas factorizations.

Mathematica

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

Prog

(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", ")))

Crossrefs

Cf. A000045, A215823, A215824, A215825.

Sequence in context: A090669 A231428 A263660 * A079374 A353579 A236217

Adjacent sequences: A215819 A215820 A215821 * A215823 A215824 A215825

Keyword

nonn,more

Author

V. Raman, Aug 23 2012

Extensions

1, 12, and 25 added by T. D. Noe, Aug 27 2012

a(29)-a(48) added by V. Raman, Aug 28 2012

Status

approved