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

1, 12, 36, 612, 684, 3852, 11628, 25308, 41004, 65484, 73188, 77292, 155268, 156636, 250308, 430236, 467172, 545148, 562428, 779076, 977364, 1244196, 1313964, 1847484, 2123028, 2185452, 2621196, 2639556, 2662812, 2707956, 2859804, 3770892, 4387428, 4679244, 4755852, 4942116, 5744916, 5795532, 6394716, 7941924, 8053308, 8270244, 9267516

Offset: 1

Author

Alexander Adamchuk, May 14 2007

Keywords

• nonn

Comments

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

Examples

			36 is a term because 36^2 = 2^4*3^4 divides 3*Fibonacci(36) = 3*14930352 = 2^4*3^4*17*19*107.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..289 (terms < 4*10^9)

Crossrefs

Cf. A000045.

Cf. A023172 (n divides Fibonacci(n)), A130163 (n^2 divides 2*Fibonacci(n)).

Programs

Magma

[n: n in [1..2*10^5] | 3*Fibonacci(n) mod n^2 eq 0 ]; // Vincenzo Librandi, Sep 17 2015

Mathematica

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 *)
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 *)
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
Length[A130164]
(* Keith Schneider, May 27 2007 *)

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",")))

Extensions

More terms from Stefan Steinerberger and Robert G. Wilson v, May 15 2007

More terms from Robert Gerbicz, Nov 28 2010