A046099 Numbers that are not cubefree. Numbers divisible by a cube greater than 1. Complement of A004709.

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 120, 125, 128, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

Comments

Also called cubeful numbers, but this term is ambiguous and is best avoided.

Numbers n such that A007427(n) = sum(d|n,mu(d)*mu(n/d)) == 0. - Benoit Cloitre, Apr 17 2002

The convention in the OEIS is that squareful, cubeful, biquadrateful (A046101), ... mean the same as "not squarefree" etc., while 2- or square-full, 3- or cube-full (A036966), 4-full (A036967) are used for Golomb's notion of powerful numbers (A001694, see references there), when each prime factor occurs to a power > 1. - M. F. Hasler, Feb 12 2008. Added by N. J. A. Sloane, Apr 25 2023: This suggestion has not been a success. It is hopeless to try to make a distinction between "cubeful" and "cubefull". To avoid ambiguity, do not use either term, but instead say exactly what you mean.

Also solutions to equation tau_{-2}(n)=0, where tau_{-2} is A007427. - Enrique Pérez Herrero, Jan 19 2013

The asymptotic density of this sequence is 1 - 1/zeta(3) = 0.168092... - Amiram Eldar, Jul 09 2020

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Cubefree.

Crossrefs

Complement of A004709.

Subsequences: A000578 and A030078.

Cf. A001694, A036966, A046101, A088453.

Programs

Haskell

a046099 n = a046099_list !! (n-1)
a046099_list = filter ((== 1) . a212793) [1..]
-- Reinhard Zumkeller, May 27 2012

Maple

isA046099 := proc(n)
    local p;
    for p in ifactors(n)[2] do
        if op(2,p) >= 3 then
            return true;
        end if;
    end do:
    false ;
end proc:
for n from 1 do
    if isA046099(n) then
        printf("%d\n",n) ;
    end if;
end do: # R. J. Mathar, Dec 08 2015

Mathematica

lst={};Do[a=0;Do[If[FactorInteger[m][[n, 2]]>2, a=1], {n, Length[FactorInteger[m]]}];If[a==1, AppendTo[lst, m]], {m, 10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 15 2008 *)

PARI

is(n)=n>7 && vecmax(factor(n)[,2])>2 \\ Charles R Greathouse IV, Sep 17 2015

Python

from sympy.ntheory.factor_ import core
def ok(n): return core(n, 3) != n
print(list(filter(ok, range(1, 337)))) # Michael S. Branicky, Aug 16 2021

Python

from sympy import mobius, integer_nthroot
def A046099(n):
    def f(x): return n+sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+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)) = 0. - Reinhard Zumkeller, May 27 2012

Sum_{n>=1} 1/a(n)^s = (zeta(s)*(zeta(3*s)-1))/zeta(3*s). - Amiram Eldar, Dec 27 2022

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Aug 15 2008

Edited by N. J. A. Sloane, Jul 27 2009