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

1,2

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

Links

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

Formula

Intersection of A000045 and A001481.

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

Example

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).

Mathematica

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

Prog

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

Crossrefs

Cf. A124132, A000045, A000161, A001481.

Sequence in context: A018559 A057196 A080637 * A007071 A242482 A085784

Adjacent sequences: A124131 A124132 A124133 * A124135 A124136 A124137

Keyword

nonn

Author

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

Extensions

More terms from Ralf Stephan, Sep 15 2013

Status

approved