A046321 Odd numbers divisible by exactly 8 primes (counted with multiplicity).

6561, 10935, 15309, 18225, 24057, 25515, 28431, 30375, 35721, 37179, 40095, 41553, 42525, 47385, 50301, 50625, 56133, 59535, 61965, 63423, 66339, 66825, 67797, 69255, 70875, 78975, 80919, 83349, 83835, 84375, 86751, 88209, 89667

Offset: 1

Author

Patrick De Geest, Jun 15 1998

Keywords

• nonn

Links

• Zak Seidov, Table of n, a(n) for n = 1..35737

Crossrefs

Cf. A046310.

Programs

Mathematica

Select[Range[1,100001,2],PrimeOmega[#]==8&] (* Harvey P. Dale, Apr 28 2018 *)

PARI

list(lim)=my(v=List()); forprime(a=3,lim\2187, my(La=lim\a); forprime(b=3,min(La\729,a), my(Lb=La\b); forprime(c=3,min(Lb\243,b), my(Lc=Lb\c); forprime(d=3,min(Lc\81,c), my(Ld=Lc\d); forprime(e=3,min(Ld\27,d), my(Le=Ld\e,E=a*b*c*d*e); forprime(f=3,min(Le\9,e), my(Lf=Le\f,F=E*f); forprime(g=3,min(Lf\3,f), my(Lg=Lf\g,G=F*g); forprime(h=3,min(Lg,g), listput(v,G*h))))))))); Set(v) \\ Charles R Greathouse IV, Aug 23 2024

Python

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A046321(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n-1+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,1,3,1,8)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Formula

a(n) ~ A046310(n) ~ 5040n log n / (log log n)^7. - Charles R Greathouse IV, Aug 23 2024

Extensions

Offset changed 0=>1 by Zak Seidov, Feb 08 2016