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

16, 24, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 81, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 225, 228, 232, 234, 240, 243

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Complement of A037144: A001222(a(n)) > 3; A117358(a(n)) > 1. - Reinhard Zumkeller, Mar 10 2006

Also numbers such that no permutation of all proper divisors exists with coprime adjacent elements: A178254(a(n)) = 0. - Reinhard Zumkeller, May 24 2010

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

Links

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

Crossrefs

Subsequence of A033942; A178212 is a subsequence.

Cf. A014613, A001055, A033273.

Programs

Maple

with(numtheory): A033987:=n->`if`(bigomega(n)>3, n, NULL): seq(A033987(n), n=1..300); # Wesley Ivan Hurt, May 26 2015

Mathematica

Select[Range[300],PrimeOmega[#]>3&] (* Harvey P. Dale, Mar 20 2016 *)

PARI

is(n)=bigomega(n)>3 \\ Charles R Greathouse IV, May 26 2015

Python

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A033987(n):
    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)))
    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)))
    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

Product p_i^e_i with Sum e_i >= 4.

A001055(a(n)) > A033273(a(n)). - Juri-Stepan Gerasimov, Nov 09 2009

a(n) ~ n. - Charles R Greathouse IV, Jul 11 2024

Extensions

More terms from Patrick De Geest, Jun 15 1998