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square-free number

Synonym: 
square free number, square free, square-free, squarefree
Type of Math Object: 
Definition
Major Section: 
Reference

Mathematics Subject Classification

11A51 no label found

Comments

Let s(n) denote the function that takes positive integers as its input and yields the number of (positive) squarefree numbers less than or equal to its input as its output. Is there a function like s(n) that already exists? Is there a general formula for s(n)? What is the limit as n approaches infinity of s(n)/n?

Wkbj79

Sorry. I should have looked up asymptotic density before posting. That answered most of my questions.

Wkbj79

In the very last line of the string of equalities, there are two equals signs. I think it would look better if the last statement (the result you're trying to prove) were on a line of its own.

Oops! This was supposed to be a correction. I've filed it properly. :-)

From this post, I realized that
the number of square-free integers smaller than x approaches
6x/pi^2, but what about the number of square-free integers
smaller than x that is "DIVISIBLE" by some prime number pi
smaller than x?
Is there a way of approximating it?
(For instance, is there a way of approximating the number of
square-free integers smaller than 10^30 that are divisible by
2 or 3 or 11?)

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