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

1, 12, 24, 168, 552, 2184, 3864, 4872, 13944, 28056, 35448, 47208, 50232, 63336, 70728, 75624, 76728, 112056, 172536, 181272, 224952, 239736, 254472, 287448, 320712, 364728, 381432, 404376, 457608, 460824, 508872, 529368, 537096, 613704, 645288, 813624

Offset

1,2

Comments

A subset of A023172.

All listed terms for n>2 are divisible by a(3) = 24 = 2^3*3.

All listed terms for n>3, except a(5), are divisible by a(4) = 168 = 2^3*3*7.

Links

Keith Schneider and Giovanni Resta, Table of n, a(n) for n = 1..776 (terms < 4*10^9, first 39 terms from Keith Schneider)

Example

24 is a term because 24^2 = 2^6*3^2 divides 2*Fibonacci(24) = 2*46368 = 2^6*3^2*7*23.

Mathematica

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 *)
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}];
A130163
Length[A130163]
(* Keith Schneider, May 27 2007 *)

Prog

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

Crossrefs

Cf. A000045.

Cf. A023172 (n divides Fibonacci(n)), A130164 (n^2 divides 3*Fibonacci(n)).

Sequence in context: A289335 A056500 A056490 * A002167 A154268 A058994

Adjacent sequences: A130160 A130161 A130162 * A130164 A130165 A130166

Keyword

nonn

Author

Alexander Adamchuk, May 14 2007

Extensions

More terms from Stefan Steinerberger, May 15 2007

a(14) corrected by N. J. A. Sloane, Nov 23 2007

Status

approved