A046309 Numbers that are divisible by at least 8 primes (counted with multiplicity).

256, 384, 512, 576, 640, 768, 864, 896, 960, 1024, 1152, 1280, 1296, 1344, 1408, 1440, 1536, 1600, 1664, 1728, 1792, 1920, 1944, 2016, 2048, 2112, 2160, 2176, 2240, 2304, 2400, 2432, 2496, 2560, 2592, 2688, 2816, 2880, 2916, 2944, 3024, 3072, 3136

Offset: 1

Author

Patrick De Geest, Jun 15 1998

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A046310.

Programs

Mathematica

Select[Range[3200],PrimeOmega[#]>7&] (* Harvey P. Dale, May 29 2013 *)

PARI

is(n)=bigomega(n)>7 \\ Charles R Greathouse IV, Sep 17 2015

Python

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A046309(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+primepi(x)+sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(2,8)))
    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

Product p_i^e_i with Sum e_i >= 8.