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

%I A052287 #35 Aug 08 2021 11:23:51
%S A052287 3,6,9,12,18,24,27,30,36,45,48,54,60,63,72,81,84,90,96,108,120,126,
%T A052287 132,135,144,150,162,168,180,189,192,198,210,216,225,234,240,243,252,
%U A052287 264,270,288,297,300,306,312,315,324,330,336,351,360,378,384,390,396
%N A052287 Start with 3; the general rule is "if x is present then so is x*y for every y <= x".
%H A052287 T. D. Noe, <a href="/A052287/b052287.txt">Table of n, a(n) for n = 1..1000</a>
%F A052287 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.
%F A052287 a(n) = 3 * A196149(n). - _Reinhard Zumkeller_, Sep 28 2011
%F A052287 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
%e A052287 63 is an element because 63 = 3*3*7 and 3 <= 3 and 7 <= 3*3.
%p A052287 N:= 1000: # get all terms <= N
%p A052287 S:= {3}:
%p A052287 New:= {3}:
%p A052287 while New <> {} do
%p A052287   x:= New[1];
%p A052287   New:= subsop(1=NULL,New);
%p A052287   R:= {seq(k*x, k=1..min(x,N/x))} minus S;
%p A052287   S:= S union R;
%p A052287   New:= New union R;
%p A052287 od:
%p A052287 sort(convert(S,list));  # _Robert Israel_, Aug 27 2015
%t A052287 3 Select[Range[132], Max[#[[2]]/#[[1]] & /@ Partition[Divisors[#], 2, 1]] <= 3 &] (* _Michael De Vlieger_, Aug 27 2015, after _Harvey P. Dale_ at A196149 *)
%o A052287 (Haskell)
%o A052287 import Data.List.Ordered (union)
%o A052287 a052287 n = a052287_list !! (n-1)
%o A052287 a052287_list = f [3] where
%o A052287    f (x:xs) = x : f (xs `union` map (x *) [2..x])
%o A052287 -- _Reinhard Zumkeller_, Jun 25 2015, Sep 28 2011
%Y A052287 If instead we start with 2, we obtain the "Nullwertzahlen sequence" A047836.
%Y A052287 Cf. A196149.
%K A052287 easy,nice,nonn
%O A052287 1,1
%A A052287 _Giuseppe Melfi_, Feb 08 2000
%E A052287 More terms from _Reinhard Zumkeller_, Jun 22 2003