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

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

Offset: 1

Author

Jeff Burch

Keywords

• nonn

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

• T. D. Noe, Table of n, a(n) for n = 1..1000

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'.

Programs

Haskell

a033942 n = a033942_list !! (n-1)
a033942_list = filter ((> 2) . a001222) [1..]
-- Reinhard Zumkeller, Oct 27 2011

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 *)

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

Python

from math import isqrt
from sympy import primepi, primerange
def A033942(n):
    def f(x): return int(n+primepi(x)+sum(primepi(x//k)-a for a,k in enumerate(primerange(isqrt(x)+1))))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

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

Extensions

Corrected by Patrick De Geest, Jun 15 1998