A160112 -id:A160112 - cp's OEIS frontend

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

Steven Finch, Jun 14 1998

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)

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.

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.

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

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:

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

{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 */

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

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

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

A160113 Number of cubefree integers not exceeding 2^n.

Original entry on oeis.org

1, 2, 4, 7, 14, 27, 54, 107, 214, 427, 854, 1706, 3410, 6815, 13629, 27259, 54521, 109042, 218080, 436158, 872318, 1744638, 3489278, 6978546, 13957092, 27914186, 55828364, 111656716, 223313428, 446626866, 893253744, 1786507472, 3573014938, 7146029910, 14292059832

Offset: 0

Gerard P. Michon, May 02 2009

An alternate definition specifying "less than 2^n" would yield the same sequence except for the first 3 terms: 0,1,3,7,14,27,54,107, etc. (since powers of 2 beyond 8 are not cubefree).

The limit of a(n)/2^n is the inverse of Apery's constant, 1/zeta(3) [see A088453].

			a(0)=1 because there is just one cubefree integer (1) not exceeding 2^0 = 1.
a(3)=7 because 1,2,3,4,5,6,7 are cubefree but 8 is not.

Chai Wah Wu, Table of n, a(n) for n = 0..103 (terms 0..80 from Gerard P. Michon)
Gerard P. Michon, On the number of cubefree integers not exceeding N.

Cf. A004709 (cubefree numbers), A160112 (decimal counterpart for cubefree integers), A143658 (binary counterpart for squarefree integers), A071172 & A053462 (decimal counterpart for squarefree integers).

Cf. A060431.

a160113 = a060431 . (2 ^)  -- Reinhard Zumkeller, Jul 27 2015

a[n_] := Sum[ MoebiusMu[i]*Floor[2^n/i^3], {i, 1, 2^(n/3)}]; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Dec 20 2011, from formula *)
Module[{nn=20,mu},mu=Table[If[Max[FactorInteger[n][[All,2]]]<3,1,0],{n,2^nn}];Table[Total[Take[mu,2^k]],{k,0,nn}]] (* The program generates the first 20 terms of the sequence. To get more, increase the value (constant) for nn, but the program may take a long time to run. *) (* Harvey P. Dale, Aug 13 2021 *)

from sympy import mobius, integer_nthroot
def A160113(n): return sum(mobius(k)*((1<Chai Wah Wu, Aug 06 2024

from bitarray import bitarray
from sympy import integer_nthroot
def A160113(n): # faster program
    q = 1<Chai Wah Wu, Aug 06 2024

a(n) = Sum_{i=1..2^(n/3)} A008683(i)*floor(2^n/i^3).

A160115 Fluctuations of the number of cubefree integers not exceeding 2^n.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 3, 0, -1, -1, 1, 2, 0, -1, 0, 2, 6, 1, 2, 7, 5, -1, -7, -4, 4, -7, -21, -7, -2, 30, 2, 14, -8, 7, -1, -7, -12, -1, 21, 28, 7, -29, -33, -76, -88, 15, 47, 58, -51, -112, 293, 122, 316, -96, -42, -259, 140, -111, 6, -790, -342, 146, 395, 1087

Offset: 0

Gerard P. Michon, May 06 2009

The asymptotic density of cubefree integers is the reciprocal of Apery's constant 1/zeta(3) = 0.83190737258... The number of cubefree integers not exceeding N is thus roughly N/zeta(3). When N is a power of 2, this sequence gives the difference between the actual number (A160113) and that linear estimate (rounded to the nearest integer).

Chai Wah Wu, Table of n, a(n) for n = 0..99
G. P. Michon, Reciprocal of Apery's constant.
G. P. Michon, On the number of cubefree integers not exceeding N.
Eric Weisstein's World of Mathematics, Cubefree.

A004709 (cubefree integers). A160112 & A160113 (counting cubefree integers).

a(n) = A160113(n)-round(2^n/zeta(3))

A160114 Fluctuations of the number of cubefree integers not exceeding 10^n.

Original entry on oeis.org

0, 1, 2, 1, 0, -1, 3, 7, -10, -1, -7, -14, -59, 21, 34, -15, 103, -104, 302, -38, -514, -290, 1130, 504, 2466, 6813, -1854, 1590, -4879, 3963, -4767, -22709

Offset: 0

Gerard P. Michon, May 06 2009

The asymptotic density of cubefree integers is the reciprocal of Apery's constant 1/zeta(3) = 0.83190737258... The number of cubefree integers not exceeding N is thus roughly N/zeta(3). When N is a power of 10, this sequence gives the difference between the actual number (A160112) and that linear estimate (rounded to the nearest integer).

G. P. Michon, Reciprocal of Apery's constant.
G. P. Michon, On the number of cubefree integers not exceeding N.
Eric Weisstein's World of Mathematics, Cubefree.

A004709 (cubefree integers). A160112 & A160113 (counting cubefree integers).

a(n) = A160112(n)-round(10^n/zeta(3))

a(30)-a(31) from Chai Wah Wu, Aug 08 2024

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

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A160113 Number of cubefree integers not exceeding 2^n.

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A160115 Fluctuations of the number of cubefree integers not exceeding 2^n.

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A160114 Fluctuations of the number of cubefree integers not exceeding 10^n.

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