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

%I A258574 #41 May 22 2025 10:21:43
%S A258574 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,
%T A258574 39,40,42,43,45,46,48,51,52,54,57,58,60,61,63,64,66,67,69,70,72,73,75,
%U A258574 76,78,79,81,82,84,85,87,88,91,93,94,96,97,100
%N A258574 Numbers n such that Fibonacci(n)+Lucas(n) is squarefree.
%C A258574 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?
%C A258574 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
%C A258574 Numbers n such that Fibonacci(n+1) is odd and squarefree. - _Chai Wah Wu_, Jun 04 2015
%C A258574 Is it a theorem that this is a subsequence of A032766? - _N. J. A. Sloane_, Jun 04 2015
%C A258574 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
%H A258574 Chai Wah Wu, <a href="/A258574/b258574.txt">Table of n, a(n) for n = 1..611</a> (based on A037918)
%t A258574 Select[Range[0, 200], SquareFreeQ[Fibonacci[#] + LucasL[#]] &]
%o A258574 (Magma) [n: n in [0..200] | IsSquarefree(Fibonacci(n)+Lucas(n))];
%o A258574 (Python)
%o A258574 from sympy import factorint
%o A258574 A258574_list = []
%o A258574 a, b = 0, 2
%o A258574 for n in range(10**2):
%o A258574     if max(factorint(b).values()) <= 1:
%o A258574         A258574_list.append(n)
%o A258574     a, b = b, a + b # _Chai Wah Wu_, Jun 04 2015
%o A258574 (PARI) is(n)=n%3<2 && issquarefree(fibonacci(n+1)) \\ _Charles R Greathouse IV_, Jun 04 2015
%o A258574 (Sage) [n for n in (0..110) if is_squarefree(2*fibonacci(n+1))] # _Bruno Berselli_,
%Y A258574 Cf. A000032, A000045, A005117, A032766, A037918, A118658.
%K A258574 nonn
%O A258574 1,3
%A A258574 _Vincenzo Librandi_, Jun 01 2015
%E A258574 Edited by _N. J. A. Sloane_, Jun 04 2015