A124134 Positive integers n such that Fibonacci(n) = a^2 + b^2, where a, b are integers.

1, 2, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 21, 23, 25, 26, 27, 29, 31, 33, 35, 37, 38, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 62, 63, 65, 67, 69, 71, 73, 74, 75, 77, 79, 81, 83, 85, 86, 87, 89, 91, 93, 95, 97, 98, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 122, 123, 125, 127

Offset: 1

Author

Melvin J. Knight (melknightdr(AT)verizon.net), Nov 30 2006

Keywords

• nonn

Comments

All odd numbers are in this sequence, since the Fibonacci number with index 2m+1 is the sum of the squares of the two Fibonacci numbers with indices m and m+1. Those with even indices ultimately depend on certain Lucas numbers being the sum of two squares (see A124132). Joint work with Kevin O'Bryant and Dennis Eichhorn.

Numbers n such that Fibonacci(n) or Fibonacci(n)/2 is a square are only 0, 1, 2, 3, 6, 12. So a and b must be distinct and nonzero for all values of this sequence except 1, 2, 3, 6, 12. - Altug Alkan, May 04 2016

Examples

			14 is in the sequence because F_14=377=11^2+16^2.
16 is not in the sequence because F_16=987 is congruent to 3 (mod 4).
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..1322 (terms 1..210 from Joerg Arndt)

Crossrefs

Cf. A124132, A000045, A000161, A001481.

Programs

Haskell

a124134 n = a124134_list !! (n-1)
a124134_list = filter ((> 0) . a000161 . a000045) [1..]
-- Reinhard Zumkeller, Oct 10 2013

Mathematica

Select[Range@ 128, SquaresR[2, Fibonacci@ #] > 0 &] (* Michael De Vlieger, May 04 2016 *)

PARI

for(n=1, 10^6, t=fibonacci(n); s=sqrtint(t); forstep(i=s, 1, -1, if(issquare(t-i*i), print1(n, ", "); break))) \\ Ralf Stephan, Sep 15 2013

PARI

is2s(n)={my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); 1; } \\ see A001481
for(n=1, 10^6, if(is2s(fibonacci(n)), print1(n, ", "))); \\ Joerg Arndt, Sep 15 2013

Python

from itertools import count, islice
from sympy import factorint, fibonacci
def A124134_gen(): # generator of terms
    return filter(lambda n:n & 1 or all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(fibonacci(n)).items()),count(1))
A124134_list = list(islice(A124134_gen(),30)) # Chai Wah Wu, Jun 27 2022

Formula

Intersection of A000045 and A001481.

A000161(A000045(a(n))) > 0. - Reinhard Zumkeller, Oct 10 2013

Extensions

More terms from Ralf Stephan, Sep 15 2013