A052287 Start with 3; the general rule is "if x is present then so is x*y for every y <= x".

3, 6, 9, 12, 18, 24, 27, 30, 36, 45, 48, 54, 60, 63, 72, 81, 84, 90, 96, 108, 120, 126, 132, 135, 144, 150, 162, 168, 180, 189, 192, 198, 210, 216, 225, 234, 240, 243, 252, 264, 270, 288, 297, 300, 306, 312, 315, 324, 330, 336, 351, 360, 378, 384, 390, 396

Offset: 1

Author

Giuseppe Melfi, Feb 08 2000

Keywords

• easy
• nice
• nonn

Examples

			63 is an element because 63 = 3*3*7 and 3 <= 3 and 7 <= 3*3.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

If instead we start with 2, we obtain the "Nullwertzahlen sequence" A047836.

Cf. A196149.

Programs

Haskell

import Data.List.Ordered (union)
a052287 n = a052287_list !! (n-1)
a052287_list = f [3] where
   f (x:xs) = x : f (xs `union` map (x *) [2..x])
-- Reinhard Zumkeller, Jun 25 2015, Sep 28 2011

Maple

N:= 1000: # get all terms <= N
S:= {3}:
New:= {3}:
while New <> {} do
  x:= New[1];
  New:= subsop(1=NULL,New);
  R:= {seq(k*x, k=1..min(x,N/x))} minus S;
  S:= S union R;
  New:= New union R;
od:
sort(convert(S,list));  # Robert Israel, Aug 27 2015

Mathematica

3 Select[Range[132], Max[#[[2]]/#[[1]] & /@ Partition[Divisors[#], 2, 1]] <= 3 &] (* Michael De Vlieger, Aug 27 2015, after Harvey P. Dale at A196149 *)

Formula

x is a term if and only if x = 3*p1*p2*...*pk with primes 2 <= p1 <= p2 <= ... <= pk and 3*p1*p2*...*pi >= p(i+1) for all i < k.

a(n) = 3 * A196149(n). - Reinhard Zumkeller, Sep 28 2011

The number of terms <= x is c*x/log(x) + O(x/(log(x))^2), where c = 0.68514..., and a(n) = C*n*log(n*log(n)) + O(n), where C = 1/c = 1.45954... This follows from the formula just above. - Andreas Weingartner, Jun 30 2021

Extensions

More terms from Reinhard Zumkeller, Jun 22 2003