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A033987 Numbers that are divisible by at least 4 primes (counted with multiplicity).

%I A033987 #35 Aug 23 2024 20:09:41
%S A033987 16,24,32,36,40,48,54,56,60,64,72,80,81,84,88,90,96,100,104,108,112,
%T A033987 120,126,128,132,135,136,140,144,150,152,156,160,162,168,176,180,184,
%U A033987 189,192,196,198,200,204,208,210,216,220,224,225,228,232,234,240,243
%N A033987 Numbers that are divisible by at least 4 primes (counted with multiplicity).
%C A033987 Complement of A037144: A001222(a(n)) > 3; A117358(a(n)) > 1. - _Reinhard Zumkeller_, Mar 10 2006
%C A033987 Also numbers such that no permutation of all proper divisors exists with coprime adjacent elements: A178254(a(n)) = 0. - _Reinhard Zumkeller_, May 24 2010
%C A033987 Also, numbers that can be written as a product of at least two composites, i.e., admit a nontrivial factorization into composites. - _Felix Fröhlich_, Dec 22 2018
%H A033987 T. D. Noe, <a href="/A033987/b033987.txt">Table of n, a(n) for n=1..1000</a>
%F A033987 Product p_i^e_i with Sum e_i >= 4.
%F A033987 A001055(a(n)) > A033273(a(n)). - _Juri-Stepan Gerasimov_, Nov 09 2009
%F A033987 a(n) ~ n. - _Charles R Greathouse IV_, Jul 11 2024
%p A033987 with(numtheory): A033987:=n->`if`(bigomega(n)>3, n, NULL): seq(A033987(n), n=1..300); # _Wesley Ivan Hurt_, May 26 2015
%t A033987 Select[Range[300],PrimeOmega[#]>3&] (* _Harvey P. Dale_, Mar 20 2016 *)
%o A033987 (PARI) is(n)=bigomega(n)>3 \\ _Charles R Greathouse IV_, May 26 2015
%o A033987 (Python)
%o A033987 from math import prod, isqrt
%o A033987 from sympy import primerange, integer_nthroot, primepi
%o A033987 def A033987(n):
%o A033987     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 A033987     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(2,4)))
%o A033987     kmin, kmax = 1,2
%o A033987     while f(kmax) >= kmax:
%o A033987         kmax <<= 1
%o A033987     while True:
%o A033987         kmid = kmax+kmin>>1
%o A033987         if f(kmid) < kmid:
%o A033987             kmax = kmid
%o A033987         else:
%o A033987             kmin = kmid
%o A033987         if kmax-kmin <= 1:
%o A033987             break
%o A033987     return kmax # _Chai Wah Wu_, Aug 23 2024
%Y A033987 Subsequence of A033942; A178212 is a subsequence.
%Y A033987 Cf. A014613, A001055, A033273.
%K A033987 nonn,easy
%O A033987 1,1
%A A033987 _N. J. A. Sloane_
%E A033987 More terms from _Patrick De Geest_, Jun 15 1998