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.

18, 24, 90, 120, 150, 180, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 6930, 9240, 11550, 13860, 16170, 18480, 20790, 23100, 25410, 27720, 90090, 120120, 150150, 180180, 210210, 240240, 270270, 300300, 330330, 360360, 390390, 420420, 450450, 480480, 1531530

Offset: 1

Author

Michael De Vlieger, Jan 28 2024

Keywords

• nonn
• easy

Comments

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.

Definition implies the following:

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

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

Superset of A369541.

Examples

			Seen as a table T(n,k) of rows n where P(n) | T(n,k)
2:    18,   24;
3:    90,  120,   150,   180;
4:   630,  840,  1050,  1260,  1470,  1680,  1890,  2100;
5:  6930, 9240, 11550, 13860, 16170, 18480, 20790, 23100, 25410, 27720;
     ...
12 is not in the sequence since 3 <= 12/6 < 5 is false.
18 is in the sequence since 3 <= 18/6 < 5 is true.
36 is not in the sequence since 3 <= 36/6 < 5 is false.
Generally, 2*P(i) is not in the sequence since 3 <= 2*P(i)/P(i) < prime(i+1) is false.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

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

Formula

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

Intersection of A364998 and A055932.

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