A046309 - OEIS

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

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OFFSET

1,1

LINKS

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

FORMULA

Product p_i^e_i with Sum e_i >= 8.

MATHEMATICA

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

PROG

(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