A082996 a(n) = card{ x <= n : bigomega(x) = 4 }.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Author

Benoit Cloitre, May 30 2003

Keywords

• nonn
• easy

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A072000, A072114.

Partial sums of A101637.

Programs

PARI

a(n)=sum(i=1,n,bigomega(i)==4)

PARI

a(n)=my(j,s);forprime(p=2,(n+.5)^(1/4),forprime(q=p,(n/p+.5)^(1/3),j=primepi(q)-2;forprime(r=q,sqrtint(n\(p*q)),s+=primepi(n\(p*q*r))-j++)));s \\ Charles R Greathouse IV, Mar 21 2012

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A082996(n): return int(sum(primepi(n//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(n,4)[0]+1)) for b,m in enumerate(primerange(k,integer_nthroot(n//k,3)[0]+1),a) for c,r in enumerate(primerange(m,isqrt(n//(k*m))+1),b))) # Chai Wah Wu, Mar 29 2025

Formula

a(n) ~ (1/6)*(n/log(n))*log(log(n))^3.