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A014613 Numbers that are products of 4 primes.

%I A014613 #61 Feb 16 2025 08:32:33
%S A014613 16,24,36,40,54,56,60,81,84,88,90,100,104,126,132,135,136,140,150,152,
%T A014613 156,184,189,196,198,204,210,220,225,228,232,234,248,250,260,276,294,
%U A014613 296,297,306,308,315,328,330,340,342,344,348,350,351,364,372,375,376
%N A014613 Numbers that are products of 4 primes.
%H A014613 T. D. Noe, <a href="/A014613/b014613.txt">Table of n, a(n) for n = 1..10000</a>
%H A014613 J. H. Conway, Heiko Dietrich and E. A. O'Brien, <a href="http://www.math.auckland.ac.nz/~obrien/research/gnu.pdf">Counting groups: gnus, moas and other exotica</a>, Math. Intell., Vol. 30, No. 2, Spring 2008.
%H A014613 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.
%F A014613 Product p_i^e_i with Sum e_i = 4.
%F A014613 a(n) ~ 6n log n / (log log n)^3. - _Charles R Greathouse IV_, May 04 2013
%F A014613 a(n) = A078840(4,n). - _R. J. Mathar_, Jan 30 2019
%t A014613 Select[Range[200], Plus @@ Last /@ FactorInteger[ # ] == 4 &] (* _Vladimir Joseph Stephan Orlovsky_, Apr 23 2008 *)
%t A014613 Select[Range[400], PrimeOmega[#] == 4&] (* _Jean-François Alcover_, Jan 17 2014 *)
%o A014613 (PARI) isA014613(n) = bigomega(n)==4 \\ _Michael B. Porter_, Dec 13 2009
%o A014613 (Python)
%o A014613 from sympy import factorint
%o A014613 def ok(n): return sum(factorint(n).values()) == 4
%o A014613 print([k for k in range(377) if ok(k)]) # _Michael S. Branicky_, Nov 19 2021
%o A014613 (Python)
%o A014613 from math import isqrt
%o A014613 from sympy import primepi, primerange, integer_nthroot
%o A014613 def A014613(n):
%o A014613     def f(x): return int(n+x-sum(primepi(x//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(x,4)[0]+1)) for b,m in enumerate(primerange(k,integer_nthroot(x//k,3)[0]+1),a) for c,r in enumerate(primerange(m,isqrt(x//(k*m))+1),b)))
%o A014613     m, k = n, f(n)
%o A014613     while m != k:
%o A014613         m, k = k, f(k)
%o A014613     return m # _Chai Wah Wu_, Aug 17 2024
%Y A014613 Cf. A033987, A114106 (number of 4-almost primes <= 10^n).
%Y A014613 Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), this sequence (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - _Jason Kimberley_, Oct 02 2011
%K A014613 nonn
%O A014613 1,1
%A A014613 _Eric W. Weisstein_
%E A014613 More terms from _Patrick De Geest_, Jun 15 1998