A356809 Fibonacci numbers which are not the sum of two squares.

3, 21, 55, 987, 2584, 6765, 17711, 46368, 317811, 832040, 2178309, 5702887, 14930352, 102334155, 267914296, 701408733, 1836311903, 4807526976, 12586269025, 32951280099, 86267571272, 225851433717, 591286729879, 1548008755920, 10610209857723

Offset: 1

Author

Ctibor O. Zizka, Aug 29 2022

Keywords

• nonn

Examples

			F(4) = 3; 3 != x^2 + y^2 as no positive integers x, y >= 0 are the solution of this Diophantine equation.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..472

Crossrefs

Intersection of A000045 and A022544.

Cf. A001481, A022340, A124132, A124134, A236264.

Programs

Mathematica

Select[Fibonacci[Range[65]], SquaresR[2, #] == 0 &] (* Amiram Eldar, Aug 29 2022 *)

PARI

is(n)=if(n%4==3, return(1)); my(f=factor(n)); for(i=1, #f~, if(f[i, 1]%4==3 && f[i, 2]%2, return(1))); 0; \\ A022544
lista(nn) = select(is, apply(fibonacci, [1..nn])); \\ Michel Marcus, Sep 04 2022

Python

from itertools import islice
from sympy import factorint
def A356809_gen(): # generator of terms
    a, b = 1, 2
    while True:
        if any(p&3==3 and e&1 for p, e in factorint(a).items()):
            yield a
        a, b = b, a+b
A356809_list = list(islice(A356809_gen(),30)) # Chai Wah Wu, Jan 10 2023