A160113 -id:A160113 - 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

A160112 Number of cubefree integers not exceeding 10^n.

Original entry on oeis.org

1, 9, 85, 833, 8319, 83190, 831910, 8319081, 83190727, 831907372, 8319073719, 83190737244, 831907372522, 8319073725828, 83190737258105, 831907372580692, 8319073725807178, 83190737258070643, 831907372580707771

Offset: 0

Gerard P. Michon, May 02 2009, May 06 2009

An alternate definition specifying "less than 10^n" would yield the same sequence except for the first 3 terms: 0, 8, 84, 833, 8319, etc. (since powers of 10 beyond 1000 are not cubefree anyhow).

The limit of a(n)/10^n is the inverse of Apery's constant, 1/zeta(3), whose digits are given by A088453.

			a(0)=1 because 1 <= 10^0 is not a multiple of the cube of a prime.
a(1)=9 because the 9 numbers 1,2,3,4,5,6,7,9,10 are cubefree; 8 is not.
a(2)=85 because there are 85 cubefree integers equal to 100 or less.
a(3)=833 because there are 833 cubefree integers below 1000 (which is not cubefree itself).

Chai Wah Wu, Table of n, a(n) for n = 0..31 (terms 0..29 from Gerard P. Michon)
Gerard P. Michon, On the number of cubefree integers not exceeding N.

A004709 (cubefree numbers). A088453 (limit of the string of digits). A160113 (binary counterpart for cubefree integers). A071172 & A053462 (decimal counterpart for squarefree integers). A143658 (binary counterpart for squarefree integers).

with(numtheory): A160112:=n->add(mobius(i)*floor(10^n/(i^3)), i=1..10^n): (1,9,85,seq(A160112(n), n=3..5)); # Wesley Ivan Hurt, Aug 01 2015

Table[ Sum[ MoebiusMu[x]*Floor[10^n/(x^3)], {x, 10^(n/3)}], {n, 0, 18}] (* Robert G. Wilson v, May 27 2009 *)

from sympy import mobius, integer_nthroot
def A160112(n): return sum(mobius(k)*(10**n//k**3) for k in range(1, integer_nthroot(10**n,3)[0]+1)) # Chai Wah Wu, Aug 06 2024

from bitarray import bitarray
from sympy import integer_nthroot
def A160112(n): # faster program
    q = 10**n
    m = integer_nthroot(q,3)[0]+1
    a, b = bitarray(m), bitarray(m)
    a[1], p, i, c = 1, 2, 4, q-sum(q//k**3 for k in range(2,m))
    while i < m:
        j = 2
        while i < m:
            if j==p:
                c -= (b[i]^1 if a[i] else -1)*(q//i**3)
                j, a[i], b[i] = 0, 1, 1
            else:
                t1, t2 = a[i], b[i]
                if (t1&t2)^1:
                    a[i], b[i] = (t1^1)&t2, ((t1^1)&t2)^1
                    c += (t2 if t1 else 2)*(q//i**3) if (t1^1)&t2 else (t2-2 if t1 else 0)*(q//i**3)
            i += p
            j += 1
        p += 1
        while a[p]|b[p]:
            p += 1
        i = p<<1
    return c # Chai Wah Wu, Aug 06 2024

a(n) = Sum_{i=1..floor(10^(n/3))} A008683(i)*floor(10^n/i^3).

A191666 Dispersion of A042964 (numbers congruent to 2 or 3 mod 4), by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 7, 5, 6, 14, 10, 8, 11, 27, 19, 15, 9, 22, 54, 38, 30, 18, 12, 43, 107, 75, 59, 35, 23, 13, 86, 214, 150, 118, 70, 46, 26, 16, 171, 427, 299, 235, 139, 91, 51, 31, 17, 342, 854, 598, 470, 278, 182, 102, 62, 34, 20, 683, 1707, 1195, 939, 555, 363

Offset: 1

Clark Kimberling, Jun 11 2011

Row 1:  A005578
Row 2:  A160113
For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion.  Each complement (beginning with its first term >1) also generates a dispersion.  The six sequences and dispersions are listed here:
...
A191663=dispersion of A042948 (0 or 1 mod 4 and >1)
A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)
A191664=dispersion of A014601 (0 or 3 mod 4 and >1)
A191665=dispersion of A042963 (1 or 2 mod 4 and >1)
A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)
A191666=dispersion of A042964 (2 or 3 mod 4)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191663 has 1st col A042964, all else A042948
A054582 has 1st col A005408, all else A005843
A191664 has 1st col A042963, all else A014601
A191665 has 1st col A014601, all else A042963
A191448 has 1st col A005843, all else A005408
A191666 has 1st col A042948, all else A042964
...
There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
   a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
   a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.

			Northwest corner:
1...2...3....6...11
4...7...14....27...54
5...10...19...38...75
8...15..30...59...118
8...18..35...70...139

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A042963, A014601, A191426, A191663.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a = 2; b = 3; m[n_] := If[Mod[n, 2] == 0, 1, 0];
f[n_] := a*m[n + 1] + b*m[n] + 4*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}]  (* A042964: (2+4k,3+4k) *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191666 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191666  *)

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

A171231 a(n) = (10*2^n + 3 - (-1)^n)/6.

Original entry on oeis.org

2, 4, 7, 14, 27, 54, 107, 214, 427, 854, 1707, 3414, 6827, 13654, 27307, 54614, 109227, 218454, 436907, 873814, 1747627, 3495254, 6990507, 13981014, 27962027, 55924054, 111848107, 223696214, 447392427, 894784854, 1789569707

Offset: 0

Paul Curtz, Dec 05 2009

From 14, the last 2 digits are of period 4: repeat [14, 27, 54, 07]. - Paul Curtz, Nov 22 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A000035, A000975, A048573, A136412 (1st bisection), 2*A136412 (2nd bisection).

Cf. A010712, A160113.

[( 10*2^n+3-(-1)^n )/6: n in [0..40]]; // Vincenzo Librandi, Aug 05 2011

LinearRecurrence[{2,1,-2},{2,4,7},40] (* Harvey P. Dale, Feb 11 2015 *)

a(n)=(10<Charles R Greathouse IV, Jul 07 2011

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), n >= 3.
a(n+1) - a(n) = A048573(n-1).
a(n) = 2*A000975(n+1) - 3*A000975(n-1).
a(n) - a(n-2) = 5*2^n.
a(n+1) - 2*a(n) = ((-1)^n-1)/2 = -A000035(n).
G.f.: ( 2-3*x^2 ) / ( (x-1)*(2*x-1)*(1+x) ). - R. J. Mathar, Jul 07 2011
a(n) = ceiling( (5/3)*(2^n) ). - Wesley Ivan Hurt, Jun 28 2013

Definition replaced by the Lava formula of 2009. Contents converted to formulas. - R. J. Mathar, Jul 07 2011

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Python

Formula

A160112 Number of cubefree integers not exceeding 10^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Python

Formula

A191666 Dispersion of A042964 (numbers congruent to 2 or 3 mod 4), by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A171231 a(n) = (10*2^n + 3 - (-1)^n)/6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions