A067259 - OEIS (original) (raw)

A067259

Cubefree numbers which are not squarefree.

36

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

COMMENTS

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

LINKS

Eric Weisstein's World of Mathematics, Cubefree

Eric Weisstein's World of Mathematics, Squarefree

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 *)

PROG

(Haskell)

a067259 n = a067259_list !! (n-1)

a067259_list = filter ((== 2) . a051903) [1..]

(Python)

from math import isqrt

from sympy import mobius, integer_nthroot

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)

EXTENSIONS

Unrelated comment removed by Jason Yuen, Apr 04 2025