A046313 Numbers that are divisible by at least 10 primes (counted with multiplicity).

1024, 1536, 2048, 2304, 2560, 3072, 3456, 3584, 3840, 4096, 4608, 5120, 5184, 5376, 5632, 5760, 6144, 6400, 6656, 6912, 7168, 7680, 7776, 8064, 8192, 8448, 8640, 8704, 8960, 9216, 9600, 9728, 9984, 10240, 10368, 10752, 11264, 11520, 11664, 11776

Offset: 1

Author

Patrick De Geest, Jun 15 1998

Keywords

• nonn

Links

• John Cerkan, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A033987, A046304, A046305, A046307, A046309, and A046311.

Cf. A046314.

Programs

Mathematica

Select[Range[12000],PrimeOmega[#]>9&] (* Harvey P. Dale, Dec 17 2018 *)

PARI

is(n)=bigomega(n)>9 \\ Charles R Greathouse IV, Sep 17 2015

Python

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A046313(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(2,10)))
    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 >= 10.

a(n) = n + O(n (log log n)^8/log n). - Charles R Greathouse IV, Apr 07 2017