A124134 -id:A124134 - cp's OEIS frontend

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

Ctibor O. Zizka, Aug 29 2022

nonn

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

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

Intersection of A000045 and A022544.

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

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

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

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

A229139 Smallest m such that Fibonacci(2n-1) = m^2 + k^2.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 9, 21, 34, 55, 89, 73, 13, 377, 610, 987, 64, 244, 4155, 4554, 10946, 2191, 28657, 15857, 74957, 34022, 29811, 50481, 134104, 832040, 162589, 387938, 711703, 1556305, 6229800, 4173137, 4059539, 1972951, 51797450, 4866315, 165580141, 46049477, 202620393, 348451533, 181781990

Offset: 1

Ralf Stephan, Sep 15 2013

nonn

Every odd-indexed Fibonacci number (A000045) is a sum of two squares (see A124134).

Which of the a(n) are not Fibonacci numbers?

			A000045(2*6-1) = 89 = 5^2 + 8^2 so a(6)=5.
A000045(2*8-1) = 610 = 9^2 + 23^2 = 13^2 + 21^2, so a(8)=9.

Cf. A000045.

Cf. A000196, A037213.

a229139 1 = 0
a229139 n = head $
   dropWhile (== 0) $ map (a037213 . (t -) . (^ 2)) [s, s - 1 ..]
   where t = a000045 (2 * n - 1); s = a000196 t
-- Reinhard Zumkeller, Oct 11 2013

for(n=1, 10^6, t=fibonacci(2*n-1);s=sqrtint(t);forstep(i=s,1,-1,if(issquare(t-i*i),print1(sqrtint(t-i*i), ",");break)))

cp's OEIS Frontend

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

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