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

1,1

Comments

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

Let us introduce a function D(n)=sigma_0(n)/(2^(alpha(1)+...+alpha(r)), sigma_0(n) number of divisors of n (A000005), prime factorization of n=p(1)^alpha(1) * ... * p(r)^alpha(r), alpha(1)+...+alpha(r) is sequence (A086436). This function splits the set of positive integers into subsets, according to the value of D(n). Squarefree numbers (005117) has D(n)=1, other numbers are "deviated" from the squarefree ideal and have 0 < D(n) < 1. So for D(n)=1/2 we have A048109, D(n)=3/4 we have A067295. - Ctibor O. Zizka, Sep 21 2008

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

Formula

A212793(a(n)) * (1 - A008966(a(n))) = 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 *)

Prog

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

Crossrefs

Cf. A004709, A005117, A059956, A088453.

Sequence in context: A304365 A038109 A375031 * A349931 A060687 A286228

Adjacent sequences: A067256 A067257 A067258 * A067260 A067261 A067262

Keyword

nonn

Author

Reinhard Zumkeller, Feb 20 2002

Status

approved