A067259 Cubefree numbers which are not squarefree.

4, 9, 12, 18, 20, 25, 28, 36, 44, 45, 49, 50, 52, 60, 63, 68, 75, 76, 84, 90, 92, 98, 99, 100, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 180, 188, 196, 198, 204, 207, 212, 220, 225, 228

Offset: 1

Author

Reinhard Zumkeller, Feb 20 2002

Keywords

• nonn

Comments

a(n)=m iff A051903(m)=2.

The asymptotic density of this sequence is 1/zeta(3) - 1/zeta(2) = A088453 - A059956 = 0.22398... - Amiram Eldar, Jul 09 2020

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Cubefree
• Eric Weisstein's World of Mathematics, Squarefree

Crossrefs

Cf. A004709, A005117, A059956, A088453.

Programs

Haskell

a067259 n = a067259_list !! (n-1)
a067259_list = filter ((== 2) . a051903) [1..]
-- Reinhard Zumkeller, May 27 2012

Mathematica

f[n_]:=Union[Last/@FactorInteger[n]][[ -1]]; lst={}; Do[If[f[n]==2,AppendTo[lst,n]],{n,2,6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 12 2010 *)
Select[Range[500], Not[SquareFreeQ[#]] && FreeQ[FactorInteger[#], {,k /;k>2}]&] (* Vaclav Kotesovec, Jul 09 2020 *)

PARI

is(n)=n>3 && vecmax(factor(n)[,2])==2 \\ Charles R Greathouse IV, Oct 15 2015

Python

from math import isqrt
from sympy import mobius, integer_nthroot
def A067259(n):
    def f(x): return n+x+sum(mobius(k)*(x//k**2-x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))+sum(mobius(k)*(x//k**2) for k in range(integer_nthroot(x,3)[0]+1,isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 05 2024

Formula

A212793(a(n)) * (1 - A008966(a(n))) = 1. - Reinhard Zumkeller, May 27 2012

Extensions

Unrelated comment removed by Jason Yuen, Apr 04 2025