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Positive integers with at least 3 prime factors (counted with multiplicity).
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%I #80 Aug 24 2024 05:57:32

%S 8,12,16,18,20,24,27,28,30,32,36,40,42,44,45,48,50,52,54,56,60,63,64,

%T 66,68,70,72,75,76,78,80,81,84,88,90,92,96,98,99,100,102,104,105,108,

%U 110,112,114,116,117,120,124,125,126,128,130,132,135,136,138,140,144

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

%C 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 A001222(a(n)) > 2; A054576(a(n)) > 1. - _Reinhard Zumkeller_, Mar 10 2006

%C Also numbers such that no permutation of all divisors exists with coprime adjacent elements: A109810(a(n))=0. - _Reinhard Zumkeller_, May 24 2010

%C A211110(a(n)) > 3. - _Reinhard Zumkeller_, Apr 02 2012

%C A060278(a(n)) > 0. - _Reinhard Zumkeller_, Apr 05 2013

%C Volumes of rectangular cuboids with each side > 1. - _Peter Woodward_, Jun 16 2015

%C If k is a term then so is k*m for m > 0. - _David A. Corneth_, Sep 30 2020

%C 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 T. D. Noe, <a href="/A033942/b033942.txt">Table of n, a(n) for n = 1..1000</a>

%F Numbers of the form Product p_i^e_i with Sum e_i >= 3.

%F a(n) ~ n. - _Charles R Greathouse IV_, May 04 2013

%p with(numtheory): A033942:=n->`if`(bigomega(n)>2, n, NULL): seq(A033942(n), n=1..200); # _Wesley Ivan Hurt_, Jun 23 2015

%t Select[ Range[150], Plus @@ Last /@ FactorInteger[ # ] > 2 &] (* _Robert G. Wilson v_, Oct 12 2005 *)

%t Select[Range[150],PrimeOmega[#]>2&] (* _Harvey P. Dale_, Jun 22 2011 *)

%o (Haskell)

%o a033942 n = a033942_list !! (n-1)

%o a033942_list = filter ((> 2) . a001222) [1..]

%o -- _Reinhard Zumkeller_, Oct 27 2011

%o (PARI) is(n)=bigomega(n)>2 \\ _Charles R Greathouse IV_, May 04 2013

%o (Python)

%o from sympy import factorint

%o def ok(n): return sum(factorint(n).values()) > 2

%o print([k for k in range(145) if ok(k)]) # _Michael S. Branicky_, Sep 10 2022

%o (Python)

%o from math import isqrt

%o from sympy import primepi, primerange

%o def A033942(n):

%o def f(x): return int(n+primepi(x)+sum(primepi(x//k)-a for a,k in enumerate(primerange(isqrt(x)+1))))

%o kmin, kmax = 1,2

%o while f(kmax) >= kmax:

%o kmax <<= 1

%o while True:

%o kmid = kmax+kmin>>1

%o if f(kmid) < kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o if kmax-kmin <= 1:

%o break

%o return kmax # _Chai Wah Wu_, Aug 23 2024

%Y Cf. A014612.

%Y A101040(a(n))=0.

%Y A033987 is a subsequence; complement of A037143. - _Reinhard Zumkeller_, May 24 2010

%Y Subsequence of A080257.

%Y See also A002808 for 'Composite numbers' or 'Divisible by at least 2 primes'.

%K nonn

%O 1,1

%A _Jeff Burch_

%E Corrected by _Patrick De Geest_, Jun 15 1998