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A033942 Positive integers with at least 3 prime factors (counted with multiplicity).

%I A033942 #80 Aug 24 2024 05:57:32
%S A033942 8,12,16,18,20,24,27,28,30,32,36,40,42,44,45,48,50,52,54,56,60,63,64,
%T A033942 66,68,70,72,75,76,78,80,81,84,88,90,92,96,98,99,100,102,104,105,108,
%U A033942 110,112,114,116,117,120,124,125,126,128,130,132,135,136,138,140,144
%N A033942 Positive integers with at least 3 prime factors (counted with multiplicity).
%C A033942 A001055(a(n)) > 2; e.g., for a(3)=18 there are 4 factorizations: 1*18 = 2*9 = 2*3*3 = 3*6. - _Reinhard Zumkeller_, Dec 29 2001
%C A033942 A001222(a(n)) > 2; A054576(a(n)) > 1. - _Reinhard Zumkeller_, Mar 10 2006
%C A033942 Also numbers such that no permutation of all divisors exists with coprime adjacent elements: A109810(a(n))=0. - _Reinhard Zumkeller_, May 24 2010
%C A033942 A211110(a(n)) > 3. - _Reinhard Zumkeller_, Apr 02 2012
%C A033942 A060278(a(n)) > 0. - _Reinhard Zumkeller_, Apr 05 2013
%C A033942 Volumes of rectangular cuboids with each side > 1. - _Peter Woodward_, Jun 16 2015
%C A033942 If k is a term then so is k*m for m > 0. - _David A. Corneth_, Sep 30 2020
%C A033942 Numbers k with a pair of proper divisors of k, (d1,d2), such that d1 < d2 and gcd(d1,d2) > 1. - _Wesley Ivan Hurt_, Jan 01 2021
%H A033942 T. D. Noe, <a href="/A033942/b033942.txt">Table of n, a(n) for n = 1..1000</a>
%F A033942 Numbers of the form Product p_i^e_i with Sum e_i >= 3.
%F A033942 a(n) ~ n. - _Charles R Greathouse IV_, May 04 2013
%p A033942 with(numtheory): A033942:=n->`if`(bigomega(n)>2, n, NULL): seq(A033942(n), n=1..200); # _Wesley Ivan Hurt_, Jun 23 2015
%t A033942 Select[ Range[150], Plus @@ Last /@ FactorInteger[ # ] > 2 &] (* _Robert G. Wilson v_, Oct 12 2005 *)
%t A033942 Select[Range[150],PrimeOmega[#]>2&] (* _Harvey P. Dale_, Jun 22 2011 *)
%o A033942 (Haskell)
%o A033942 a033942 n = a033942_list !! (n-1)
%o A033942 a033942_list = filter ((> 2) . a001222) [1..]
%o A033942 -- _Reinhard Zumkeller_, Oct 27 2011
%o A033942 (PARI) is(n)=bigomega(n)>2 \\ _Charles R Greathouse IV_, May 04 2013
%o A033942 (Python)
%o A033942 from sympy import factorint
%o A033942 def ok(n): return sum(factorint(n).values()) > 2
%o A033942 print([k for k in range(145) if ok(k)]) # _Michael S. Branicky_, Sep 10 2022
%o A033942 (Python)
%o A033942 from math import isqrt
%o A033942 from sympy import primepi, primerange
%o A033942 def A033942(n):
%o A033942     def f(x): return int(n+primepi(x)+sum(primepi(x//k)-a for a,k in enumerate(primerange(isqrt(x)+1))))
%o A033942     kmin, kmax = 1,2
%o A033942     while f(kmax) >= kmax:
%o A033942         kmax <<= 1
%o A033942     while True:
%o A033942         kmid = kmax+kmin>>1
%o A033942         if f(kmid) < kmid:
%o A033942             kmax = kmid
%o A033942         else:
%o A033942             kmin = kmid
%o A033942         if kmax-kmin <= 1:
%o A033942             break
%o A033942     return kmax # _Chai Wah Wu_, Aug 23 2024
%Y A033942 Cf. A014612.
%Y A033942 A101040(a(n))=0.
%Y A033942 A033987 is a subsequence; complement of A037143. - _Reinhard Zumkeller_, May 24 2010
%Y A033942 Subsequence of A080257.
%Y A033942 See also A002808 for 'Composite numbers' or 'Divisible by at least 2 primes'.
%K A033942 nonn
%O A033942 1,1
%A A033942 _Jeff Burch_
%E A033942 Corrected by _Patrick De Geest_, Jun 15 1998