A375055 Nonsquarefree numbers k divisible by at least 3 distinct primes.

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 396, 408, 414, 420, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492, 495

Offset: 1

Author

Michael De Vlieger, Oct 22 2024

Keywords

• nonn
• easy

Comments

Also, numbers k such that there exists a pair of necessarily composite divisors {d, k/d}, d < k/d, with quality Q, i.e., gcd(d, k/d) > 1 but there exists a prime p | d that does not divide k/d, and also a prime q | k/d that does not divide d.

A178212 is a proper subset.

This sequence is distinct from A123712 since 420 is here.

This sequence is distinct from A182855 since 360 is here.

Examples

			a(1) = 60 = 2^2 * 3 * 5, the smallest number such that bigomega(60) > omega(60) > 2. Bigomega(60) = 4, omega(60) = 3.
72 is not in the sequence because it is the product of 2 distinct prime factors.
a(2) = 84 = 2^2 * 3 * 7, since bigomega(84) = 4, omega(84) = 3.
a(3) = 90 = 2 * 3^2 * 5, since bigomega(90) = 4, omega(90) = 3.
a(4) = 120 = 2^3 * 3 * 5, since bigomega(120) = 5, omega(120) = 3.
210 is not in the sequence because it is squarefree.
a(35) = 360 = 2^3 * 3^2 * 5 since bigomega(360) = 6, omega(360) = 3.
a(43) = 420 = 2^2 * 3 * 5 * 7 since bigomega(420) = 5, omega(420) = 4, etc.
.
Table showing pairs of factors of a(n) for select n, such that the pair possesses quality Q (see comments).
    n    a(n)   pair of factors with quality Q.
  -------------------------------------------------------------------
    1     60    6 X 10;
    2     84    6 X 14;
    3     90    6 X 15;
    4    120    6 X 20,  10 X 12;
    5    126    6 X 21;
    6    132    6 X 22;
    7    140   10 X 14;
    8    150   10 X 15;
   17    240    6 X 40,  10 X 24, 12 X 20;
   51    480    6 X 80,  10 X 48, 12 X 40, 20 X 24;
  117    840    6 X 140, 10 X 84, 12 X 70, 14 X 60, 20 X 42, 28 X 30.
		

Links

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

Crossrefs

Cf. A001221, A001222, A013929, A024619, A126706, A178212.

Programs

Mathematica

Select[Range[500], PrimeOmega[#] > PrimeNu[#] > 2 &]

Formula

{a(n)} = { k : bigomega(k) > omega(k) > 2 }, where bigomega = A001222 and omega = A001221.