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A033942 Positive integers with at least 3 prime factors (counted with multiplicity). 35
8, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 125, 126, 128, 130, 132, 135, 136, 138, 140, 144 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
A001222(a(n)) > 2; A054576(a(n)) > 1. - Reinhard Zumkeller, Mar 10 2006
Also numbers such that no permutation of all divisors exists with coprime adjacent elements: A109810(a(n))=0. - Reinhard Zumkeller, May 24 2010
A211110(a(n)) > 3. - Reinhard Zumkeller, Apr 02 2012
A060278(a(n)) > 0. - Reinhard Zumkeller, Apr 05 2013
Volumes of rectangular cuboids with each side > 1. - Peter Woodward, Jun 16 2015
If k is a term then so is k*m for m > 0. - David A. Corneth, Sep 30 2020
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
LINKS
FORMULA
Numbers of the form Product p_i^e_i with Sum e_i >= 3.
a(n) ~ n. - Charles R Greathouse IV, May 04 2013
MAPLE
with(numtheory): A033942:=n->`if`(bigomega(n)>2, n, NULL): seq(A033942(n), n=1..200); # Wesley Ivan Hurt, Jun 23 2015
MATHEMATICA
Select[ Range[150], Plus @@ Last /@ FactorInteger[ # ] > 2 &] (* Robert G. Wilson v, Oct 12 2005 *)
Select[Range[150], PrimeOmega[#]>2&] (* Harvey P. Dale, Jun 22 2011 *)
PROG
(Haskell)
a033942 n = a033942_list !! (n-1)
a033942_list = filter ((> 2) . a001222) [1..]
-- Reinhard Zumkeller, Oct 27 2011
(PARI) is(n)=bigomega(n)>2 \\ Charles R Greathouse IV, May 04 2013
(Python)
from sympy import factorint
def ok(n): return sum(factorint(n).values()) > 2
print([k for k in range(145) if ok(k)]) # Michael S. Branicky, Sep 10 2022
CROSSREFS
Cf. A014612.
A101040(a(n))=0.
A033987 is a subsequence; complement of A037143. - Reinhard Zumkeller, May 24 2010
Subsequence of A080257.
See also A002808 for 'Composite numbers' or 'Divisible by at least 2 primes'.
Sequence in context: A152758 A160258 A071280 * A111087 A341610 A337373
KEYWORD
nonn
AUTHOR
EXTENSIONS
Corrected by Patrick De Geest, Jun 15 1998
STATUS
approved

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Last modified March 19 07:31 EDT 2024. Contains 370955 sequences. (Running on oeis4.)