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

%I A046311 #22 Sep 10 2024 00:23:40
%S A046311 512,768,1024,1152,1280,1536,1728,1792,1920,2048,2304,2560,2592,2688,
%T A046311 2816,2880,3072,3200,3328,3456,3584,3840,3888,4032,4096,4224,4320,
%U A046311 4352,4480,4608,4800,4864,4992,5120,5184,5376,5632,5760,5832,5888,6048,6144
%N A046311 Numbers that are divisible by at least 9 primes (counted with multiplicity).
%H A046311 John Cerkan, <a href="/A046311/b046311.txt">Table of n, a(n) for n = 1..10000</a>
%F A046311 Product p_i^e_i with Sum e_i >= 9.
%F A046311 a(n) = n + O(n (log log n)^7/log n). - _Charles R Greathouse IV_, Apr 07 2017
%t A046311 Select[Range[6200],PrimeOmega[#]>8&] (* _Harvey P. Dale_, May 20 2013 *)
%o A046311 (PARI) is(n)=bigomega(n)>8 \\ _Charles R Greathouse IV_, Sep 17 2015
%o A046311 (Python)
%o A046311 from math import isqrt, prod
%o A046311 from sympy import primerange, integer_nthroot, primepi
%o A046311 def A046311(n):
%o A046311     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)))
%o A046311     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)))
%o A046311     def bisection(f,kmin=0,kmax=1):
%o A046311         while f(kmax) > kmax: kmax <<= 1
%o A046311         while kmax-kmin > 1:
%o A046311             kmid = kmax+kmin>>1
%o A046311             if f(kmid) <= kmid:
%o A046311                 kmax = kmid
%o A046311             else:
%o A046311                 kmin = kmid
%o A046311         return kmax
%o A046311     return bisection(f,n,n) # _Chai Wah Wu_, Sep 09 2024
%Y A046311 Subsequence of A033987, A046304, A046305, A046307, and A046309.
%Y A046311 Cf. A046312.
%K A046311 nonn
%O A046311 1,1
%A A046311 _Patrick De Geest_, Jun 15 1998