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A046309 Numbers that are divisible by at least 8 primes (counted with multiplicity).

%I A046309 #15 Aug 23 2024 22:07:43
%S A046309 256,384,512,576,640,768,864,896,960,1024,1152,1280,1296,1344,1408,
%T A046309 1440,1536,1600,1664,1728,1792,1920,1944,2016,2048,2112,2160,2176,
%U A046309 2240,2304,2400,2432,2496,2560,2592,2688,2816,2880,2916,2944,3024,3072,3136
%N A046309 Numbers that are divisible by at least 8 primes (counted with multiplicity).
%H A046309 Harvey P. Dale, <a href="/A046309/b046309.txt">Table of n, a(n) for n = 1..1000</a>
%F A046309 Product p_i^e_i with Sum e_i >= 8.
%t A046309 Select[Range[3200],PrimeOmega[#]>7&] (* _Harvey P. Dale_, May 29 2013 *)
%o A046309 (PARI) is(n)=bigomega(n)>7 \\ _Charles R Greathouse IV_, Sep 17 2015
%o A046309 (Python)
%o A046309 from math import prod, isqrt
%o A046309 from sympy import primerange, integer_nthroot, primepi
%o A046309 def A046309(n):
%o A046309     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 A046309     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)))
%o A046309     kmin, kmax = 1,2
%o A046309     while f(kmax) >= kmax:
%o A046309         kmax <<= 1
%o A046309     while True:
%o A046309         kmid = kmax+kmin>>1
%o A046309         if f(kmid) < kmid:
%o A046309             kmax = kmid
%o A046309         else:
%o A046309             kmin = kmid
%o A046309         if kmax-kmin <= 1:
%o A046309             break
%o A046309     return kmax # _Chai Wah Wu_, Aug 23 2024
%Y A046309 Cf. A046310.
%K A046309 nonn
%O A046309 1,1
%A A046309 _Patrick De Geest_, Jun 15 1998