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A037917 Numbers n such that the Fibonacci number F(n) is divisible by a square.

%I A037917 #44 Feb 16 2025 08:32:37
%S A037917 6,12,18,24,25,30,36,42,48,50,54,56,60,66,72,75,78,84,90,91,96,100,
%T A037917 102,108,110,112,114,120,125,126,132,138,144,150,153,156,162,168,174,
%U A037917 175,180,182,186,192,198,200,204,210,216,220,222,224,225,228,234,240
%N A037917 Numbers n such that the Fibonacci number F(n) is divisible by a square.
%C A037917 Is a(n) asymptotic to C*n with 4 < C < 4.5 ? - _Benoit Cloitre_, Sep 04 2002
%C A037917 Numbers are a superset of the multiples of 6 (A008588), because 8 divides Fibonacci(6m) = A134492(m). Sequence apparently also contains the multiples of 25. Are all a(n) composite? Members not divisible by 6 or 25 are 56, 91, 110, 112, 153, 182, 220, 224, 273, 280, ... - _Ralf Stephan_, Jan 26 2014
%C A037917 These numbers are the positive multiples of A065069. - _Charles R Greathouse IV_, Feb 02 2014
%C A037917 To address Cloitre's question, if such C exists it must be less than 4.3 using the known terms of A065069. - _Charles R Greathouse IV_, Feb 04 2014
%H A037917 Amiram Eldar, <a href="/A037917/b037917.txt">Table of n, a(n) for n = 1..328</a> (terms 1..235 from T. D. Noe, based on Kelly's table)
%H A037917 Blair Kelly, <a href="http://mersennus.net/fibonacci//f1000.txt">Fibonacci Factorizations</a>
%H A037917 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number.</a>
%t A037917 Select[Range[100], MoebiusMu[Fibonacci[#]] == 0 &] (* _Alonso del Arte_, Jan 26 2014 *)
%o A037917 (PARI) is(n)=!issquarefree(fibonacci(n)) \\ _Charles R Greathouse IV_, Feb 02 2014
%Y A037917 Cf. A000045, A037918, A001177, A030426.
%K A037917 nonn
%O A037917 1,1
%A A037917 _Marc LeBrun_
%E A037917 More terms from _Eric W. Weisstein_