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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.
3

%I #6 Feb 03 2024 09:56:00

%S 18,24,90,120,150,180,630,840,1050,1260,1470,1680,1890,2100,6930,9240,

%T 11550,13860,16170,18480,20790,23100,25410,27720,90090,120120,150150,

%U 180180,210210,240240,270270,300300,330330,360360,390390,420420,450450,480480,1531530

%N 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 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 Definition implies the following:

%C 1.) A119288(k) = 3 since all terms are even, hence 6 | k.

%C 2.) k is a product m * P(n), n > 1, such that rad(m) | P(n) and 3 <= m < prime(n+1).

%C Superset of A369541.

%H Michael De Vlieger, <a href="/A369540/b369540.txt">Table of n, a(n) for n = 1..10000</a>

%F {a(n)} = { m × P(n) : 3 <= m < q, n >= 2 }.

%F Intersection of A364998 and A055932.

%F A060735 without primorials P(i) and 2*P(i).

%e Seen as a table T(n,k) of rows n where P(n) | T(n,k)

%e 2: 18, 24;

%e 3: 90, 120, 150, 180;

%e 4: 630, 840, 1050, 1260, 1470, 1680, 1890, 2100;

%e 5: 6930, 9240, 11550, 13860, 16170, 18480, 20790, 23100, 25410, 27720;

%e ...

%e 12 is not in the sequence since 3 <= 12/6 < 5 is false.

%e 18 is in the sequence since 3 <= 18/6 < 5 is true.

%e 36 is not in the sequence since 3 <= 36/6 < 5 is false.

%e Generally, 2*P(i) is not in the sequence since 3 <= 2*P(i)/P(i) < prime(i+1) is false.

%t P = 2; Table[P *= Prime[n]; Array[# P &, Prime[n + 1] - 3, 3], {n, 2, 6}] // Flatten

%Y Cf. A002110, A007947, A053669, A055932, A060735, A119288, A126706, A364998, A369541.

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Jan 28 2024