OFFSET
1,2
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)
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
GĂ©rard 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.
FORMULA
A066990(a(n)) = a(n). - Reinhard Zumkeller, Jun 25 2009
A212793(a(n)) = 1. - Reinhard Zumkeller, May 27 2012
Sum_{n>=1} 1/a(n)^s = zeta(s)/zeta(3*s), for s > 1. - Amiram Eldar, Dec 27 2022
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 *)
PROG
(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 */
(Haskell)
a004709 n = a004709_list !! (n-1)
a004709_list = filter ((== 1) . a212793) [1..]
-- Reinhard Zumkeller, May 27 2012
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Steven Finch, Jun 14 1998
STATUS
approved