This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A072114 #42 Aug 17 2024 01:34:16
%S A072114 0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,4,4,4,5,6,6,7,7,7,7,
%T A072114 7,7,7,7,7,7,7,7,8,8,9,10,10,10,10,10,11,11,12,12,12,12,12,12,12,12,
%U A072114 12,12,12,13,13,13,14,14,15,15,16,16,16,16,16,17,18,18,19,19,19,19,19,19,19
%N A072114 Number of 3-almost primes (A014612) <= n.
%C A072114 Number of k <= n such that bigomega(k) = 3.
%C A072114 Let A be a positive integer then card{ x <= n : bigomega(x) = A } ~ (n/Log(n))*Log(Log(n))^(A-1)/(A-1)!. For which n, card{ x <= n : bigomega(x) = 3 } >= card{ x <= n : bigomega(x) = 2 } ?
%C A072114 15530 is the first number for which there are more 3-almost primes than 2-almost primes. See A125149.
%D A072114 E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1, Teubner, Leipzig; third edition : Chelsea, New York (1974).
%D A072114 G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.
%H A072114 Charles R Greathouse IV, <a href="/A072114/b072114.txt">Table of n, a(n) for n = 0..10000</a>
%H A072114 E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, <a href="http://name.umdl.umich.edu/ABV2766.0001.001">vol. 1</a> and <a href="http://name.umdl.umich.edu/ABV2766.0002.001">vol. 2</a>, Leipzig, Berlin, B. G. Teubner, 1909.
%F A072114 a(n) = card{ x <= n : bigomega(x) = 3 }, asymptotically : a(n) ~ (n/log(n))*log(log(n))^2/2 [Landau, p. 211].
%t A072114 Table[Sum[KroneckerDelta[PrimeOmega[i], 3], {i, n}], {n, 0, 50}] (* _Wesley Ivan Hurt_, Oct 07 2014 *)
%o A072114 (PARI) for(n=1,100,print1(sum(i=1,n,bigomega(i)==3),","))
%o A072114 (PARI) a(n)=my(j,s);forprime(p=2,(n+.5)^(1/3),j=primepi(p)-2;forprime(q=p,sqrtint(n\p),s+=primepi(n\(p*q))-j++));s \\ _Charles R Greathouse IV_, Mar 21 2012
%o A072114 (Python)
%o A072114 from math import isqrt
%o A072114 from sympy import primepi, primerange, integer_nthroot
%o A072114 def A072114(n): return int(sum(primepi(n//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(n,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(n//k)+1),a))) # _Chai Wah Wu_, Aug 17 2024
%Y A072114 Cf. A014612, A109251, A001358, A072000.
%Y A072114 Partial sums of A101605.
%Y A072114 Cf. A125149.
%K A072114 easy,nonn
%O A072114 0,13
%A A072114 _Benoit Cloitre_, Jun 19 2002