A046305 Numbers that are divisible by at least 6 primes (counted with multiplicity).

64, 96, 128, 144, 160, 192, 216, 224, 240, 256, 288, 320, 324, 336, 352, 360, 384, 400, 416, 432, 448, 480, 486, 504, 512, 528, 540, 544, 560, 576, 600, 608, 624, 640, 648, 672, 704, 720, 729, 736, 756, 768, 784, 792, 800, 810, 816, 832, 840, 864, 880, 896

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 and A046304.

Cf. A046306.

Programs

Mathematica

Select[Range[1000],Total[Transpose[FactorInteger[#]][[2]]]>5&]  (* Harvey P. Dale, Jan 13 2011 *)
Select[Range[1000],PrimeOmega[#]>5&] (* Harvey P. Dale, Apr 14 2019 *)

PARI

is(n)=bigomega(n)>5 \\ Charles R Greathouse IV, Sep 17 2015

Python

from math import prod, isqrt
from sympy import primerange, primepi, integer_nthroot
def A046305(n):
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def almostprimepi(n, k):
        if k==0: return int(n>=1)
        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)))
        return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n, 0, 1, 1, k)) if k>1 else primepi(n))
    def f(x): return n+1+sum(almostprimepi(x, k) for k in range(1, 6))
    return bisection(f, n, n) # Chai Wah Wu, Mar 29 2025

Formula

Product p_i^e_i with Sum e_i >= 6.

a(n) = n + O(n (log log n)^4/log n). - Charles R Greathouse IV, Apr 07 2017

Extensions

Offset corrected by Andrew Howroyd, Aug 13 2024