A215818 - OEIS

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A215818

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

1, 2, 3, 4, 6, 9, 11, 12, 13, 18, 23, 25, 26, 33, 36, 39, 47, 52, 59, 61, 71, 83, 107, 109, 121, 122, 131, 141, 157, 167, 179, 183, 191, 193, 227, 244, 249, 251, 299, 314, 321, 337, 359, 363, 383, 393, 397, 423, 431, 433, 471, 501, 517, 579, 587, 601, 628, 647, 649, 794, 866, 877, 911, 913, 947, 1079, 1091, 1093

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

These Fibonacci numbers F(n) have no prime factor congruent to 5 or 7 mod 8 to an odd power.

LINKS

Table of n, a(n) for n=1..68.

Blair Kelly, Fibonacci and Lucas factorizations

MATHEMATICA

Select[Range[200], Length[FindInstance[x^2 + 2 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]%8>4&&a[2, j]%2==1, has=1; break)); if(has==0, print(i", ")))