A072000 Number of semiprimes (A001358) <= n.

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, 10, 11, 12, 13, 13, 13, 14, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 20, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 26

Offset: 1

Author

Benoit Cloitre, Jun 19 2002

Keywords

• easy
• nonn

Comments

Number of k <= n such that bigomega(k) = 2.

References

• 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.
• E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1, Teubner, Leipzig, 1909; third edition : Chelsea, New York (1974).
• G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.

Links

• Daniel Forgues, Table of n, a(n) for n = 1..40882
• Dragos Crisan and Radek Erban, On the counting function of semiprimes, arXiv:2006.16491 [math.NT], 2020.
• E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.
• Eric Weisstein's World of Mathematics, Semiprime

Crossrefs

Cf. A000720, A001358, A066265, A064911.

Programs

Maple

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

Mathematica

semiPrimePi[n_] := Sum[ PrimePi[n/Prime@i] -i + 1, {i, PrimePi@Sqrt@n}]; Array[semiPrimePi, 78] (* Robert G. Wilson v, Jan 03 2006 *)
(* If version >= 7 *) a[n_] := Select[Range[n], PrimeOmega[#] == 2 &] // Length; Table[a[n], {n, 1, 77}] (* Jean-François Alcover, Jun 29 2013 *)
Accumulate[Table[If[PrimeOmega[n]==2,1,0],{n,100}]] (* Harvey P. Dale, Jun 14 2014 *)

PARI

for(n=1,100,print1(sum(i=1,n,if(bigomega(i)-2,0,1)),","))

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

Python

from math import isqrt
from sympy import primepi, prime
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

Formula

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]

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

a(n) = card { x <= n : bigomega(x) = 2 }.

Asymptotically a(n) ~ n*log(log(n))/log(n). [Landau, p. 211]

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]

a(n) = A072613(n) + A056811(n). - R. J. Mathar, Jun 10 2007

a(n) = Sum_{i=1..n} A064911(i). - Jonathan Vos Post, Dec 30 2007

a(n)*A064911(n) = A174956(n). - Reinhard Zumkeller, Apr 03 2010

Extensions

Edited by Robert G. Wilson v, Feb 15 2006