A046311 Numbers that are divisible by at least 9 primes (counted with multiplicity).

512, 768, 1024, 1152, 1280, 1536, 1728, 1792, 1920, 2048, 2304, 2560, 2592, 2688, 2816, 2880, 3072, 3200, 3328, 3456, 3584, 3840, 3888, 4032, 4096, 4224, 4320, 4352, 4480, 4608, 4800, 4864, 4992, 5120, 5184, 5376, 5632, 5760, 5832, 5888, 6048, 6144

Offset

1,1

Links

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

Formula

Product p_i^e_i with Sum e_i >= 9.

a(n) = n + O(n (log log n)^7/log n). - Charles R Greathouse IV, Apr 07 2017

Mathematica

Select[Range[6200], PrimeOmega[#]>8&] (* Harvey P. Dale, May 20 2013 *)

Prog

(PARI) is(n)=bigomega(n)>8 \\ Charles R Greathouse IV, Sep 17 2015
(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A046311(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+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, 9)))
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f, n, n) # Chai Wah Wu, Sep 09 2024

Crossrefs

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

Cf. A046312.

Sequence in context: A234136 A248536 A202461 * A036333 A046312 A045033

Adjacent sequences: A046308 A046309 A046310 * A046312 A046313 A046314

Keyword

nonn

Author

Patrick De Geest, Jun 15 1998

Status

approved