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

1,2

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

Links

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A000045.

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

Sequence in context: A085331 A225100 A058040 * A144973 A043358 A023731

Adjacent sequences: A130161 A130162 A130163 * A130165 A130166 A130167

Keyword

nonn

Author

Alexander Adamchuk, May 14 2007

Extensions

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

More terms from Robert Gerbicz, Nov 28 2010

Status

approved