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

%I A046305 #35 Apr 21 2025 23:05:12
%S A046305 64,96,128,144,160,192,216,224,240,256,288,320,324,336,352,360,384,
%T A046305 400,416,432,448,480,486,504,512,528,540,544,560,576,600,608,624,640,
%U A046305 648,672,704,720,729,736,756,768,784,792,800,810,816,832,840,864,880,896
%N A046305 Numbers that are divisible by at least 6 primes (counted with multiplicity).
%H A046305 John Cerkan, <a href="/A046305/b046305.txt">Table of n, a(n) for n = 1..10000</a>
%F A046305 Product p_i^e_i with Sum e_i >= 6.
%F A046305 a(n) = n + O(n (log log n)^4/log n). - _Charles R Greathouse IV_, Apr 07 2017
%t A046305 Select[Range[1000],Total[Transpose[FactorInteger[#]][[2]]]>5&]  (* _Harvey P. Dale_, Jan 13 2011 *)
%t A046305 Select[Range[1000],PrimeOmega[#]>5&] (* _Harvey P. Dale_, Apr 14 2019 *)
%o A046305 (PARI) is(n)=bigomega(n)>5 \\ _Charles R Greathouse IV_, Sep 17 2015
%o A046305 (Python)
%o A046305 from math import prod, isqrt
%o A046305 from sympy import primerange, primepi, integer_nthroot
%o A046305 def A046305(n):
%o A046305     def bisection(f, kmin=0, kmax=1):
%o A046305         while f(kmax) > kmax: kmax <<= 1
%o A046305         kmin = kmax >> 1
%o A046305         while kmax-kmin > 1:
%o A046305             kmid = kmax+kmin>>1
%o A046305             if f(kmid) <= kmid:
%o A046305                 kmax = kmid
%o A046305             else:
%o A046305                 kmin = kmid
%o A046305         return kmax
%o A046305     def almostprimepi(n, k):
%o A046305         if k==0: return int(n>=1)
%o A046305         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 A046305         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))
%o A046305     def f(x): return n+1+sum(almostprimepi(x, k) for k in range(1, 6))
%o A046305     return bisection(f, n, n) # _Chai Wah Wu_, Mar 29 2025
%Y A046305 Subsequence of A033987 and A046304.
%Y A046305 Cf. A046306.
%K A046305 nonn
%O A046305 1,1
%A A046305 _Patrick De Geest_, Jun 15 1998
%E A046305 Offset corrected by _Andrew Howroyd_, Aug 13 2024