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 A217856 Numbers with three prime factors, not necessarily distinct, except cubes of primes. 2

%I

%S 12,18,20,28,30,42,44,45,50,52,63,66,68,70,75,76,78,92,98,99,102,105,

%T 110,114,116,117,124,130,138,147,148,153,154,164,165,170,171,172,174,

%U 175,182,186,188,190,195,207,212,222,230,231,236,238,242,244,245,246

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

%C Union of A007304 and A054753.

%C 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.

%H Vincenzo Librandi, <a href="/A217856/b217856.txt">Table of n, a(n) for n = 1..1000</a>

%H Wushi Goldring, <a href="http://dx.doi.org/10.1016/j.jnt.2005.10.010">Dynamics of the w function and primes</a>, Journal of Number Theory, Volume 119, Issue 1, July 2006, Pages 86-98.

%e 12 = 2^2 * 3 = 2 * 2 * 3, and so it is in the sequence.

%e 27 = 3^3 = 3 * 3 * 3, but that's only one distinct prime and hence 27 is not in the sequence.

%e 30 = 2 * 3 * 5, and so it is in the sequence.

%t Select[Range[300], PrimeOmega[#] == 3 && PrimeNu[#] > 1 &] (* _Alonso del Arte_, Oct 14 2012 *)

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

%o (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,", "))););}

%Y Cf. A217857.

%K nonn,easy

%O 1,1

%A _Michel Marcus_, Oct 13 2012

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Last modified May 10 01:09 EDT 2021. Contains 343747 sequences. (Running on oeis4.)