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A046304 Divisible by at least 5 primes (counted with multiplicity).

%I A046304 #29 Apr 21 2025 23:05:08
%S A046304 32,48,64,72,80,96,108,112,120,128,144,160,162,168,176,180,192,200,
%T A046304 208,216,224,240,243,252,256,264,270,272,280,288,300,304,312,320,324,
%U A046304 336,352,360,368,378,384,392,396,400,405,408,416,420,432,440,448,450,456
%N A046304 Divisible by at least 5 primes (counted with multiplicity).
%H A046304 John Cerkan, <a href="/A046304/b046304.txt">Table of n, a(n) for n = 1..10000</a>
%F A046304 Product p_i^e_i with Sum e_i >= 5.
%F A046304 a(n) = n + O(n (log log n)^3/log n). - _Charles R Greathouse IV_, Apr 07 2017
%t A046304 Select[Range[500],PrimeOmega[#]>4&] (* _Harvey P. Dale_, Apr 16 2013 *)
%o A046304 (PARI) is(n)=bigomega(n)>4 \\ _Charles R Greathouse IV_, Sep 17 2015
%o A046304 (Python)
%o A046304 from math import prod, isqrt
%o A046304 from sympy import primerange, primepi, integer_nthroot
%o A046304 def A046304(n):
%o A046304     def bisection(f, kmin=0, kmax=1):
%o A046304         while f(kmax) > kmax: kmax <<= 1
%o A046304         kmin = kmax >> 1
%o A046304         while kmax-kmin > 1:
%o A046304             kmid = kmax+kmin>>1
%o A046304             if f(kmid) <= kmid:
%o A046304                 kmax = kmid
%o A046304             else:
%o A046304                 kmin = kmid
%o A046304         return kmax
%o A046304     def almostprimepi(n, k):
%o A046304         if k==0: return int(n>=1)
%o A046304         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 A046304         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 A046304     def f(x): return n+1+sum(almostprimepi(x,k) for k in range(1,5))
%o A046304     return bisection(f,n,n) # _Chai Wah Wu_, Mar 29 2025
%Y A046304 Subsequence of A033987.
%Y A046304 Cf. A014614.
%K A046304 nonn
%O A046304 1,1
%A A046304 _Patrick De Geest_, Jun 15 1998