%I #67 Jul 23 2024 14:57:30
%S 0,0,0,1,1,2,2,2,3,4,4,4,4,5,6,6,6,6,6,6,7,8,8,8,9,10,10,10,10,10,10,
%T 10,11,12,13,13,13,14,15,15,15,15,15,15,15,16,16,16,17,17,18,18,18,18,
%U 19,19,20,21,21,21,21,22,22,22,23,23,23,23,24,24,24,24,24,25,25,25,26
%N Number of semiprimes (A001358) <= n.
%C Number of k <= n such that bigomega(k) = 2.
%D A. Hildebrand, On the number of prime factors of an integer, in Ramanujan Revisited (Urbana-Champaign, Ill., 1987), pp. 167-185, Academic Press, Boston, MA, 1988.
%D E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1, Teubner, Leipzig, 1909; third edition : Chelsea, New York (1974).
%D G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.
%H Daniel Forgues, <a href="/A072000/b072000.txt">Table of n, a(n) for n = 1..40882</a>
%H Dragos Crisan and Radek Erban, <a href="https://arxiv.org/abs/2006.16491">On the counting function of semiprimes</a>, arXiv:2006.16491 [math.NT], 2020.
%H 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.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Semiprime.html">Semiprime</a>
%F Let PrimePi(x) denote the number of primes <= x (cf. A000720). Then 2*a(n) = Sum_{ primes p <= n/2 } PrimePi(n/p) + PrimePi(sqrt(n)). [Landau, p. 211]
%F Let PrimePi(x) denote the number of primes <= x (cf. A000720). Then a(n) = Sum_{i=1..PrimePi(sqrt(n))} (PrimePi(n/prime(i)) - i + 1). - _Robert G. Wilson v_, Feb 07 2006
%F a(n) = card { x <= n : bigomega(x) = 2 }.
%F Asymptotically a(n) ~ n*log(log(n))/log(n). [Landau, p. 211]
%F Let A be a positive integer. Then card { x <= n : bigomega(x) = A } ~ (n/log(n))*log(log(n))^(A-1)/(A-1)! [Landau, p. 211]
%F a(n) = A072613(n) + A056811(n). - _R. J. Mathar_, Jun 10 2007
%F a(n) = Sum_{i=1..n} A064911(i). - _Jonathan Vos Post_, Dec 30 2007
%F a(n)*A064911(n) = A174956(n). - _Reinhard Zumkeller_, Apr 03 2010
%p A072000 := proc(n) local sp,t ; sp := 0 ; for t from 1 to n do if numtheory[bigomega](t) = 2 then sp := sp+1 ; fi ; od ; sp ; end proc: # _R. J. Mathar_, Jun 10 2007
%t semiPrimePi[n_] := Sum[ PrimePi[n/Prime@i] -i + 1, {i, PrimePi@Sqrt@n}]; Array[semiPrimePi, 78] (* _Robert G. Wilson v_, Jan 03 2006 *)
%t (* If version >= 7 *) a[n_] := Select[Range[n], PrimeOmega[#] == 2 &] // Length; Table[a[n], {n, 1, 77}] (* _Jean-François Alcover_, Jun 29 2013 *)
%t Accumulate[Table[If[PrimeOmega[n]==2,1,0],{n,100}]] (* _Harvey P. Dale_, Jun 14 2014 *)
%o (PARI) for(n=1,100,print1(sum(i=1,n,if(bigomega(i)-2,0,1)),","))
%o (PARI) a(n)=my(s=0,i=0); forprime(p=2, sqrt(n), s+=primepi(n\p); i++); s - i * (i-1)/2 \\ _Charles R Greathouse IV_, Apr 21 2011
%o (Python)
%o from math import isqrt
%o from sympy import primepi, prime
%o def A072000(n): return int(sum(primepi(n//prime(k))-k+1 for k in range(1,primepi(isqrt(n))+1))) # _Chai Wah Wu_, Jul 23 2024
%Y Cf. A000720, A001358, A066265, A064911.
%K easy,nonn
%O 1,6
%A _Benoit Cloitre_, Jun 19 2002
%E Edited by _Robert G. Wilson v_, Feb 15 2006