A004709 Cubefree numbers: numbers that are not divisible by any cube > 1.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85

Offset: 1

Author

Steven Finch, Jun 14 1998

Keywords

• nonn
• easy

Comments

Numbers n such that no smaller number m satisfies: kronecker(n,k)=kronecker(m,k) for all k. - Michael Somos, Sep 22 2005

The asymptotic density of cubefree integers is the reciprocal of Apery's constant 1/zeta(3) = A088453. - Gerard P. Michon, May 06 2009

The Schnirelmann density of the cubefree numbers is 157/189 (Orr, 1969). - Amiram Eldar, Mar 12 2021

From Amiram Eldar, Feb 26 2024: (Start)

Numbers whose sets of unitary divisors (A077610) and bi-unitary divisors (A222266) coincide.

Number whose all divisors are (1+e)-divisors, or equivalently, numbers k such that A049599(k) = A000005(k). (End)

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Gérard P. Michon, On the number of cubefree integers not exceeding N.
• Richard C. Orr, On the Schnirelmann density of the sequence of k-free integers, Journal of the London Mathematical Society, Vol. 1, No. 1 (1969), pp. 313-319.
• Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv preprint, arXiv:1511.03860 [math.NT], 2015.
• Eric Weisstein's World of Mathematics, Cubefree.

Crossrefs

Complement of A046099.

Cf. A005117 (squarefree), A067259 (cubefree but not squarefree), A046099 (cubeful).

Cf. A160112, A160113, A160114 & A160115: On the number of cubefree integers. - Gerard P. Michon, May 06 2009

Cf. A030078.

Cf. A001221, A124010, A212793.

Cf. A000005, A049599, A077610, A222266, A376365, A376366.

Programs

Haskell

a004709 n = a004709_list !! (n-1)
a004709_list = filter ((== 1) . a212793) [1..]
-- Reinhard Zumkeller, May 27 2012

Maple

isA004709 := proc(n)
    local p;
    for p in ifactors(n)[2] do
        if op(2,p) > 2 then
            return false;
        end if;
    end do:
    true ;
end proc:

Mathematica

Select[Range[6!], FreeQ[FactorInteger[#], {, k /; k > 2}] &] (* Jan Mangaldan, May 07 2014 *)

PARI

{a(n)= local(m,c); if(n<2, n==1, c=1; m=1; while( cvecmax(factor(m)[,2]), c++)); m)} /* Michael Somos, Sep 22 2005 */

Python

from sympy.ntheory.factor_ import core
def ok(n): return core(n, 3) == n
print(list(filter(ok, range(1, 86)))) # Michael S. Branicky, Aug 16 2021

Python

from sympy import mobius, integer_nthroot
def A004709(n):
    def f(x): return n+x-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

A066990(a(n)) = a(n). - Reinhard Zumkeller, Jun 25 2009

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

A124010(a(n),k) <= 2 for all k = 1..A001221(a(n)). - Reinhard Zumkeller, Mar 04 2015

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