A336442 - OEIS

%I #9 Jul 26 2020 02:34:09

%S 60,120,140,180,210,240,280,300,315,360,420,462,480,504,540,560,600,

%T 616,630,660,693,700,720,728,770,780,792,819,840,900,910,924,936,945,

%U 960,980,990,1001,1008,1020,1050,1080,1092,1120,1140,1144,1170,1200,1232,1260

%N Numbers having 3 pairwise coprime divisors, {d_1, d_2, d_3}, such that d_1 < d_2 < d_3 < 2*d_1.

%C Erdős (1970) proved that the asymptotic density of this sequence exists and is less than 1.

%C The numbers of terms not exceeding 10^k for k = 1, 2, ... are 0, 1, 37, 543, 6529, 73578, 798916, 8480417, 88832422, ...

%C Any positive multiple of any term of this sequence is also a term. The primitive terms are in A336443.

%H Amiram Eldar, <a href="/A336442/b336442.txt">Table of n, a(n) for n = 1..10000</a>

%H Paul Erdős, <a href="https://www.renyi.hu/~p_erdos/1970-21.pdf">Some extremal problems in combinatorial number theory</a>, in the book Hari Shankar (ed.), Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio (1970), pp. 123-133.

%e 60 is a term since {3, 4, 5} are divisors of 60, gcd(3,4) = gcd(4,5) = gcd(3,5) = 1 and 3 < 4 < 5 < 2*3.

%t divQ[n_] := AnyTrue[Subsets[Divisors[n], {3}], And @@ CoprimeQ @@@ Subsets[#, {2}] && #[[3]] < 2 * #[[1]] &]; Select[Range[1500], divQ]

%Y Subsequence of A005279.

%Y A336443 is a subsequence.

%K nonn

%O 1,1

%A _Amiram Eldar_, Jul 21 2020