A160112 Number of cubefree integers not exceeding 10^n.

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

Offset

0,2

Comments

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.

Links

G. P. Michon, Table of n, a(n) for = n=0..29

G. P. Michon, On the number of cubefree integers not exceeding N.

Formula

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

Example

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).

Maple

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

Mathematica

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

Crossrefs

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).

Sequence in context: A228417 A015580 A163308 * A108427 A152106 A142982

Adjacent sequences: A160109 A160110 A160111 * A160113 A160114 A160115

Keyword

easy,nice,nonn

Author

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

Status

approved