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A130164 Numbers k such that k^2 divides 3*Fibonacci(k).

%I A130164 #26 Sep 08 2022 08:45:30
%S A130164 1,12,36,612,684,3852,11628,25308,41004,65484,73188,77292,155268,
%T A130164 156636,250308,430236,467172,545148,562428,779076,977364,1244196,
%U A130164 1313964,1847484,2123028,2185452,2621196,2639556,2662812,2707956,2859804,3770892,4387428,4679244,4755852,4942116,5744916,5795532,6394716,7941924,8053308,8270244,9267516
%N A130164 Numbers k such that k^2 divides 3*Fibonacci(k).
%C A130164 A subset of A023172. All listed terms for n>1 are divisible by a(2) = 12 = 2^2*3. All listed terms for n>2 are divisible by a(3) = 36 = 2^2*3^2. - _Robert G. Wilson v_, May 15 2007
%H A130164 Giovanni Resta, <a href="/A130164/b130164.txt">Table of n, a(n) for n = 1..289</a> (terms < 4*10^9)
%e A130164 36 is a term because 36^2 = 2^4*3^4 divides 3*Fibonacci(36) = 3*14930352 = 2^4*3^4*17*19*107.
%t A130164 a=0; b=1; c=1; Do[ a=b; b=c; c=a+b; If[ Mod[3c,(n+2)^2 ] == 0, Print[n+2]],{n, 1, 30000}] (* _Stefan Steinerberger_, May 15 2007 *)
%t A130164 a = 0; b = 0; c = 1; lst = {}; Do[ If[ Mod[3c, n^2] == 0, AppendTo[lst, n]]; a = b; b = c; c = a + b; {n, 2000000}]; lst (* _Robert G. Wilson v_ *)
%t A130164 A130164 = {1}; a = 0; b = 12; c = 3864; Do[If[Mod[36b, n^2] == 0, A130164 = Append[A130164, n]]; a = b; b = c; c = 322b - a;, {n, 12, 1000000, 12}]; A130164
%t A130164 Length[A130164]
%t A130164 (* Keith Schneider, May 27 2007 *)
%o A130164 (PARI) for(n=1,10^7,A=matrix(2,2,i,j,Mod(1,n*n)*(i+j<4))^n;if(lift(3*A[1,2])==0,print1(n",")))
%o A130164 (Magma) [n: n in [1..2*10^5] | 3*Fibonacci(n) mod n^2 eq 0 ]; // _Vincenzo Librandi_, Sep 17 2015
%Y A130164 Cf. A000045.
%Y A130164 Cf. A023172 (n divides Fibonacci(n)), A130163 (n^2 divides 2*Fibonacci(n)).
%K A130164 nonn
%O A130164 1,2
%A A130164 _Alexander Adamchuk_, May 14 2007
%E A130164 More terms from _Stefan Steinerberger_ and _Robert G. Wilson v_, May 15 2007
%E A130164 More terms from _Robert Gerbicz_, Nov 28 2010