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A130163 Numbers k such that k^2 divides 2*Fibonacci(k).

%I A130163 #29 Sep 08 2022 08:45:30
%S A130163 1,12,24,168,552,2184,3864,4872,13944,28056,35448,47208,50232,63336,
%T A130163 70728,75624,76728,112056,172536,181272,224952,239736,254472,287448,
%U A130163 320712,364728,381432,404376,457608,460824,508872,529368,537096,613704,645288,813624
%N A130163 Numbers k such that k^2 divides 2*Fibonacci(k).
%C A130163 A subset of A023172.
%C A130163 All listed terms for n>2 are divisible by a(3) = 24 = 2^3*3.
%C A130163 All listed terms for n>3, except a(5), are divisible by a(4) = 168 = 2^3*3*7.
%H A130163 Keith Schneider and Giovanni Resta, <a href="/A130163/b130163.txt">Table of n, a(n) for n = 1..776</a> (terms < 4*10^9, first 39 terms from Keith Schneider)
%e A130163 24 is a term because 24^2 = 2^6*3^2 divides 2*Fibonacci(24) = 2*46368 = 2^6*3^2*7*23.
%t A130163 a=0; b=1; c=1; Do[a=b; b=c; c=a+b; If[Mod[2c,(n+2)^2]==0,Print[n+2]],{n,1,40000}] (* _Stefan Steinerberger_, May 15 2007 *)
%t A130163 A130163 = {1}; a = 0; b = 12; c = 3864; Do[If[Mod[24b, n^2] == 0, A130163 = Append[A130163, n]]; a = b; b = c; c = 322b - a;, {n, 12, 1000000, 12}];
%t A130163 A130163
%t A130163 Length[A130163]
%t A130163 (* Keith Schneider, May 27 2007 *)
%o A130163 (Magma) [n: n in [1..2*10^5] | 2*Fibonacci(n) mod n^2 eq 0 ]; // _Vincenzo Librandi_, Sep 17 2015
%Y A130163 Cf. A000045.
%Y A130163 Cf. A023172 (n divides Fibonacci(n)), A130164 (n^2 divides 3*Fibonacci(n)).
%K A130163 nonn
%O A130163 1,2
%A A130163 _Alexander Adamchuk_, May 14 2007
%E A130163 More terms from _Stefan Steinerberger_, May 15 2007
%E A130163 a(14) corrected by _N. J. A. Sloane_, Nov 23 2007