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

%I A004709 #106 Feb 16 2025 08:32:28
%S A004709 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 A004709 30,31,33,34,35,36,37,38,39,41,42,43,44,45,46,47,49,50,51,52,53,55,57,
%U A004709 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 A004709 Cubefree numbers: numbers that are not divisible by any cube > 1.
%C A004709 Numbers n such that no smaller number m satisfies: kronecker(n,k)=kronecker(m,k) for all k. - _Michael Somos_, Sep 22 2005
%C A004709 The asymptotic density of cubefree integers is the reciprocal of Apery's constant 1/zeta(3) = A088453. - _Gerard P. Michon_, May 06 2009
%C A004709 The Schnirelmann density of the cubefree numbers is 157/189 (Orr, 1969). - _Amiram Eldar_, Mar 12 2021
%C A004709 From _Amiram Eldar_, Feb 26 2024: (Start)
%C A004709 Numbers whose sets of unitary divisors (A077610) and bi-unitary divisors (A222266) coincide.
%C A004709 Number whose all divisors are (1+e)-divisors, or equivalently, numbers k such that A049599(k) = A000005(k). (End)
%H A004709 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 A004709 Gérard P. Michon, <a href="http://www.numericana.com/answer/counting.htm#cubefree">On the number of cubefree integers not exceeding N</a>.
%H A004709 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 A004709 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 A004709 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Cubefree.html">Cubefree</a>.
%F A004709 A066990(a(n)) = a(n). - _Reinhard Zumkeller_, Jun 25 2009
%F A004709 A212793(a(n)) = 1. - _Reinhard Zumkeller_, May 27 2012
%F A004709 A124010(a(n),k) <= 2 for all k = 1..A001221(a(n)). - _Reinhard Zumkeller_, Mar 04 2015
%F A004709 Sum_{n>=1} 1/a(n)^s = zeta(s)/zeta(3*s), for s > 1. - _Amiram Eldar_, Dec 27 2022
%p A004709 isA004709 := proc(n)
%p A004709     local p;
%p A004709     for p in ifactors(n)[2] do
%p A004709         if op(2,p) > 2 then
%p A004709             return false;
%p A004709         end if;
%p A004709     end do:
%p A004709     true ;
%p A004709 end proc:
%t A004709 Select[Range[6!], FreeQ[FactorInteger[#], {_, k_ /; k > 2}] &] (* _Jan Mangaldan_, May 07 2014 *)
%o A004709 (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 A004709 (Haskell)
%o A004709 a004709 n = a004709_list !! (n-1)
%o A004709 a004709_list = filter ((== 1) . a212793) [1..]
%o A004709 -- _Reinhard Zumkeller_, May 27 2012
%o A004709 (Python)
%o A004709 from sympy.ntheory.factor_ import core
%o A004709 def ok(n): return core(n, 3) == n
%o A004709 print(list(filter(ok, range(1, 86)))) # _Michael S. Branicky_, Aug 16 2021
%o A004709 (Python)
%o A004709 from sympy import mobius, integer_nthroot
%o A004709 def A004709(n):
%o A004709     def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
%o A004709     m, k = n, f(n)
%o A004709     while m != k:
%o A004709         m, k = k, f(k)
%o A004709     return m # _Chai Wah Wu_, Aug 05 2024
%Y A004709 Complement of A046099.
%Y A004709 Cf. A005117 (squarefree), A067259 (cubefree but not squarefree), A046099 (cubeful).
%Y A004709 Cf. A160112, A160113, A160114 & A160115: On the number of cubefree integers. - _Gerard P. Michon_, May 06 2009
%Y A004709 Cf. A030078.
%Y A004709 Cf. A001221, A124010, A212793.
%Y A004709 Cf. A000005, A049599, A077610, A222266, A376365, A376366.
%K A004709 nonn,easy
%O A004709 1,2
%A A004709 _Steven Finch_, Jun 14 1998