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A375055
Nonsquarefree numbers k divisible by at least 3 distinct primes.
2
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
1,1
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.
LINKS
FORMULA
{a(n)} = { k : bigomega(k) > omega(k) > 2 }, where bigomega = A001222 and omega = A001221.
EXAMPLE
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.
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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.
MATHEMATICA
Select[Range[500], PrimeOmega[#] > PrimeNu[#] > 2 &]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Oct 22 2024
STATUS
approved