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%I #39 Sep 08 2022 08:46:12
%S 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 39,40,42,43,45,46,48,51,52,54,57,58,60,61,63,64,66,67,69,70,72,73,75,
%U 76,78,79,81,82,84,85,87,88,91,93,94,96,97,100
%N Numbers n such that Fibonacci(n)+Lucas(n) is squarefree.
%C 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 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 Numbers n such that Fibonacci(n+1) is odd and squarefree. - _Chai Wah Wu_, Jun 04 2015
%C Is it a theorem that this is a subsequence of A032766? - _N. J. A. Sloane_, Jun 04 2015
%C 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 Chai Wah Wu, <a href="/A258574/b258574.txt">Table of n, a(n) for n = 1..611</a> (based on A037918)
%t Select[Range[0, 200], SquareFreeQ[Fibonacci[#] + LucasL[#]] &]
%o (Magma) [n: n in [0..200] | IsSquarefree(Fibonacci(n)+Lucas(n))];
%o (Python)
%o from sympy import factorint
%o A258574_list = []
%o a, b = 0, 2
%o for n in range(10**2):
%o ....if max(factorint(b).values()) <= 1:
%o ........A258574_list.append(n)
%o ....a, b = b, a + b # _Chai Wah Wu_, Jun 04 2015
%o (PARI) is(n)=n%3<2 && issquarefree(fibonacci(n+1)) \\ _Charles R Greathouse IV_, Jun 04 2015
%o (Sage) [n for n in (0..110) if is_squarefree(2*fibonacci(n+1))] # _Bruno Berselli_,
%Y Cf. A000032, A000045, A005117, A032766, A037918, A118658.
%K nonn
%O 1,3
%A _Vincenzo Librandi_, Jun 01 2015
%E Edited by _N. J. A. Sloane_, Jun 04 2015
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