A258574 Numbers n such that Fibonacci(n)+Lucas(n) is squarefree.

0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 25, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 42, 43, 45, 46, 48, 51, 52, 54, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 91, 93, 94, 96, 97, 100

Offset: 1

Author

Vincenzo Librandi, Jun 01 2015

Keywords

• nonn

Comments

It appears that the sequence consists of the numbers congruent to 0 or 1 mod 3 (A032766) except for 24, 49, 55, 90, 99, 109, 111, ... What are these exceptions?

Also numbers n such that 2*Fibonacci(n+1) is squarefree because Lucas(n) = Fibonacci(n-1)+Fibonacci(n+1). - Michel Lagneau, Jun 04 2015

Numbers n such that Fibonacci(n+1) is odd and squarefree. - Chai Wah Wu, Jun 04 2015

Is it a theorem that this is a subsequence of A032766? - N. J. A. Sloane, Jun 04 2015

This sequence is a subsequence of A032766. Proof: since Fibonacci(0) = 0 and Fibonacci(1) = 1, Fibonacci(n) mod 2 has the pattern: 0, 1, 1, 0, 1, 1, 0, ..., i.e. if n mod 3 = 0, then Fibonacci(n) is even, and n-1 is not a member of this sequence. In other words, members of this sequence must be congruent to 0 or 1 mod 3. - Chai Wah Wu, Jun 04 2015

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..611 (based on A037918)

Crossrefs

Cf. A000032, A000045, A005117, A032766, A037918, A118658.

Programs

Magma

[n: n in [0..200] | IsSquarefree(Fibonacci(n)+Lucas(n))];

Mathematica

Select[Range[0, 200], SquareFreeQ[Fibonacci[#] + LucasL[#]] &]

PARI

is(n)=n%3<2 && issquarefree(fibonacci(n+1)) \\ Charles R Greathouse IV, Jun 04 2015

Python

from sympy import factorint
A258574_list = []
a, b = 0, 2
for n in range(10**2):
    if max(factorint(b).values()) <= 1:
        A258574_list.append(n)
    a, b = b, a + b # Chai Wah Wu, Jun 04 2015

Sage

[n for n in (0..110) if is_squarefree(2*fibonacci(n+1))] # Bruno Berselli,

Extensions

Edited by N. J. A. Sloane, Jun 04 2015