A263161 Positive values of n such that A000071(n+2) is divisible by A000217(n).

1, 240, 600, 768, 1008, 1200, 1320, 1800, 2160, 2688, 2736, 3000, 3360, 3888, 4800, 5280, 5520, 6120, 6479, 6480, 6720, 6840, 7320, 7680, 8208, 8640, 9000, 9600, 9720, 10368, 11160, 11663, 12240, 12288, 13200, 13248, 13440, 13680, 14400, 15120, 15360, 15456, 16560, 18048

Offset

1,2

Comments

Interestingly, the minimum value of a(n) - a(n-1) is 1. Is there a maximum value of a(n) - a(n-1)?

From Robert Israel, Oct 19 2015: (Start)

n is in the sequence if either n is odd and A001175(n) and A001175((n+1)/2) both divide n+1, or n is even and A001175(n/2) and A001175(n+1) both divide n.

Most of the terms of the sequence appear to fall in these categories. The first two that do not are 15456 and 41640.

In particular, if n = 2^j * 3^k * 5^m with j >= 4, k >= 1 and m >= 1, and n+1 is prime, then n is in the sequence. There are believed to be infinitely many numbers of this form. The first few are 240, 1200, 2160, 4800, 6480, 7680, 8640, 9600, 14400, 15360. (End)

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Example

For n = 1, A000071(1+2) = 1 is divisible by A000217(1) = 1.

Maple

fmod:= proc(a, b) local A, t;
  uses LinearAlgebra[Modular];
  if b < 4295022903 then t:= integer[8] else t:= integer fi;
  A:= Mod(b, <<1, 1>|<1, 0>>, t);
  MatrixPower(b, A, a)[1, 2];
end proc:
filter:= n -> (fmod(n+2, n*(n+1)/2) = 1):
filter(1):= true:
select(filter, [$1..10^5]); # Robert Israel, Oct 19 2015

Mathematica

fQ[n_] := Mod[Fibonacci[n + 2] - 1, n (n + 1)/2] == 0; Select[Range@20000, fQ] (* Bruno Berselli, Oct 19 2015 - after Robert G. Wilson v in A263225 *)

Prog

(PARI) for(n=1, 20000, if((fibonacci(n+2)-1) % (n*(n+1)/2) == 0, print1(n", ")));
(PARI) is(n)=((Mod([1, 1; 1, 0], n*(n+1)/2))^(n+2))[1, 2]==1 \\ Charles R Greathouse IV, Oct 19 2015
(Magma) [n: n in [1..20000] | IsDivisibleBy(Fibonacci(n+2)-1, n*(n+1) div 2)]; // Bruno Berselli, Oct 19 2015

Crossrefs

Cf. A000045, A000071, A000217, A001175, A263225.

Sequence in context: A063372 A070123 A119659 * A268772 A202196 A060663

Adjacent sequences: A263158 A263159 A263160 * A263162 A263163 A263164

Keyword

nonn

Author

Altug Alkan, Oct 11 2015

Status

approved