A217856 Numbers with three prime factors, not necessarily distinct, except cubes of primes.

12, 18, 20, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, 102, 105, 110, 114, 116, 117, 124, 130, 138, 147, 148, 153, 154, 164, 165, 170, 171, 172, 174, 175, 182, 186, 188, 190, 195, 207, 212, 222, 230, 231, 236, 238, 242, 244, 245, 246

Offset: 1

Author

Michel Marcus, Oct 13 2012

Keywords

• nonn
• easy

Comments

Union of A007304 and A054753.

If n belongs to the sequence, then it can written n=pqr where p, q, r are primes and possibly two, but not all three of them are equal. It is named A3 in the link.

Examples

			12 = 2^2 * 3 = 2 * 2 * 3, and so it is in the sequence.
27 = 3^3 = 3 * 3 * 3, but that's only one distinct prime and hence 27 is not in the sequence.
30 = 2 * 3 * 5, and so it is in the sequence.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Wushi Goldring, Dynamics of the w function and primes, Journal of Number Theory, Volume 119, Issue 1, July 2006, Pages 86-98.

Crossrefs

Cf. A217857.

Programs

Mathematica

Select[Range[300], PrimeOmega[#] == 3 && PrimeNu[#] > 1 &] (* Alonso del Arte, Oct 14 2012 *)

PARI

atr(n) = {for (i=2, n,if (bigomega(i) == 3 && omega(i) > 1, print1(i, ", ");););}

PARI

atr(n) = {for (i=2, n,f = factor(i); len = length(f~);if (len > 1,s = sum(i=1, len, f[i,2]);if (s == 3, print1(i,", "))););}