A004709 - OEIS

%I #105 Sep 22 2024 03:55:58

%S 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,

%T 30,31,33,34,35,36,37,38,39,41,42,43,44,45,46,47,49,50,51,52,53,55,57,

%U 58,59,60,61,62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,82,83,84,85

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

%C Numbers n such that no smaller number m satisfies: kronecker(n,k)=kronecker(m,k) for all k. - _Michael Somos_, Sep 22 2005

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

%C The Schnirelmann density of the cubefree numbers is 157/189 (Orr, 1969). - _Amiram Eldar_, Mar 12 2021

%C From _Amiram Eldar_, Feb 26 2024: (Start)

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

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

%H Amiram Eldar, <a href="/A004709/b004709.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)

%H Gérard P. Michon, <a href="http://www.numericana.com/answer/counting.htm#cubefree">On the number of cubefree integers not exceeding N</a>.

%H Richard C. Orr, <a href="https://doi.org/10.1112/jlms/s1-44.1.313">On the Schnirelmann density of the sequence of k-free integers</a>, Journal of the London Mathematical Society, Vol. 1, No. 1 (1969), pp. 313-319.

%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1511.03860">Set of all densities of exponentially S-numbers</a>, arXiv preprint, arXiv:1511.03860 [math.NT], 2015.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cubefree.html">Cubefree</a>.

%F A066990(a(n)) = a(n). - _Reinhard Zumkeller_, Jun 25 2009

%F A212793(a(n)) = 1. - _Reinhard Zumkeller_, May 27 2012

%F A124010(a(n),k) <= 2 for all k = 1..A001221(a(n)). - _Reinhard Zumkeller_, Mar 04 2015

%F Sum_{n>=1} 1/a(n)^s = zeta(s)/zeta(3*s), for s > 1. - _Amiram Eldar_, Dec 27 2022

%p isA004709 := proc(n)

%p     local p;

%p     for p in ifactors(n)[2] do

%p         if op(2,p) > 2 then

%p             return false;

%p         end if;

%p     end do:

%p     true ;

%p end proc:

%t Select[Range[6!], FreeQ[FactorInteger[#], {_, k_ /; k > 2}] &] (* _Jan Mangaldan_, May 07 2014 *)

%o (PARI) {a(n)= local(m,c); if(n<2, n==1, c=1; m=1; while( c<n, m++; if( 3>vecmax(factor(m)[,2]), c++)); m)} /* _Michael Somos_, Sep 22 2005 */

%o (Haskell)

%o a004709 n = a004709_list !! (n-1)

%o a004709_list = filter ((== 1) . a212793) [1..]

%o -- _Reinhard Zumkeller_, May 27 2012

%o (Python)

%o from sympy.ntheory.factor_ import core

%o def ok(n): return core(n, 3) == n

%o print(list(filter(ok, range(1, 86)))) # _Michael S. Branicky_, Aug 16 2021

%o (Python)

%o from sympy import mobius, integer_nthroot

%o def A004709(n):

%o     def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))

%o     m, k = n, f(n)

%o     while m != k:

%o         m, k = k, f(k)

%o     return m # _Chai Wah Wu_, Aug 05 2024

%Y Complement of A046099.

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

%Y Cf. A160112, A160113, A160114 & A160115: On the number of cubefree integers. - _Gerard P. Michon_, May 06 2009

%Y Cf. A030078.

%Y Cf. A001221, A124010, A212793.

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

%K nonn,easy

%O 1,2

%A _Steven Finch_, Jun 14 1998