cp's OEIS Frontend

A369540 Numbers k neither squarefree nor prime powers such that A119288(k) <= k/A007947(k) < A053669(k) and A007947(k) is a primorial P(i) = A002110(i) for some i.

%I A369540 #6 Feb 03 2024 09:56:00
%S A369540 18,24,90,120,150,180,630,840,1050,1260,1470,1680,1890,2100,6930,9240,
%T A369540 11550,13860,16170,18480,20790,23100,25410,27720,90090,120120,150150,
%U A369540 180180,210210,240240,270270,300300,330330,360360,390390,420420,450450,480480,1531530
%N A369540 Numbers k neither squarefree nor prime powers such that A119288(k) <= k/A007947(k) < A053669(k) and A007947(k) is a primorial P(i) = A002110(i) for some i.
%C A369540 Nonsquarefree numbers k such that omega(k) > 1, whose squarefree kernel rad(k) is a primorial, with second least prime factor not greater than k/rad(k), and k/rad(k) is smaller than the smallest nondivisor prime.
%C A369540 Definition implies the following:
%C A369540 1.) A119288(k) = 3 since all terms are even, hence 6 | k.
%C A369540 2.) k is a product m * P(n), n > 1, such that rad(m) | P(n) and 3 <= m < prime(n+1).
%C A369540 Superset of A369541.
%H A369540 Michael De Vlieger, <a href="/A369540/b369540.txt">Table of n, a(n) for n = 1..10000</a>
%F A369540 {a(n)} = { m × P(n) : 3 <= m < q, n >= 2 }.
%F A369540 Intersection of A364998 and A055932.
%F A369540 A060735 without primorials P(i) and 2*P(i).
%e A369540 Seen as a table T(n,k) of rows n where P(n) | T(n,k)
%e A369540 2:    18,   24;
%e A369540 3:    90,  120,   150,   180;
%e A369540 4:   630,  840,  1050,  1260,  1470,  1680,  1890,  2100;
%e A369540 5:  6930, 9240, 11550, 13860, 16170, 18480, 20790, 23100, 25410, 27720;
%e A369540      ...
%e A369540 12 is not in the sequence since 3 <= 12/6 < 5 is false.
%e A369540 18 is in the sequence since 3 <= 18/6 < 5 is true.
%e A369540 36 is not in the sequence since 3 <= 36/6 < 5 is false.
%e A369540 Generally, 2*P(i) is not in the sequence since 3 <= 2*P(i)/P(i) < prime(i+1) is false.
%t A369540 P = 2; Table[P *= Prime[n]; Array[# P &, Prime[n + 1] - 3, 3], {n, 2, 6}] // Flatten
%Y A369540 Cf. A002110, A007947, A053669, A055932, A060735, A119288, A126706, A364998, A369541.
%K A369540 nonn,easy
%O A369540 1,1
%A A369540 _Michael De Vlieger_, Jan 28 2024