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

1,1

Links

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

Formula

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

Mathematica

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

Prog

(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
(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

Crossrefs

Cf. A046310.

Sequence in context: A137231 A321824 A043488 * A043456 A255114 A255629

Adjacent sequences: A046318 A046319 A046320 * A046322 A046323 A046324

Keyword

nonn

Author

Patrick De Geest, Jun 15 1998

Extensions

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

Status

approved