# asymtotic ratio of nonsquarefree repunits

Let $R_n:=\frac{10^n-1}{10-1}$ (called a repunit) and $\mu$ be the Moebius function. Also $[n]:=\{1,2,3,\cdots, n\}, A_n:=\{m \in [n]| \mu (R_m)=0\}.$

What is the value of $\lim \limits_{n \rightarrow \infty} \frac{|A_n|}{n}?$

Find the values of that for $\mu(R_n)=1$ and $\mu(R_n)=-1.$

I computed several values using PIE.

$r_1=0.111111\cdots$, $r_2=0.151515\cdots$, $r_3=0.165945165945\cdots$, $r_4=0.1726517265\cdots$

Then $\lim \limits_{n \rightarrow \infty} \frac{|A_n|}{n}=\lim \limits_{n \rightarrow \infty} r_n.$

First minimal nonsquarefree repunit is $R_9$. So, $r_1= \frac{1}{9}$. Second minimal nonsquarefree repunit $R_n$ where $n$ is not multiple of 9 is $R_{22}$. So, $r_2= \frac{1}{9}+\frac{1}{22}-\frac{1}{198},$ etc.

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You have not defined these $r_i$. – Gerry Myerson Jul 11 '12 at 7:00
You still haven't defined the $r_i$, at any rate I can't extract a definition from your calculation of the first two values. – Gerry Myerson Jul 11 '12 at 7:31
Next one is $R_{42}$. So $r_3 = \frac{1}{9} + \frac{1}{22} + \frac{1}{42} - \frac{1}{(9,22)} - \frac{1}{(9,42)} - \frac{1}{(22,42)} + \frac{1}{(9,22,42)},$ where $(a,b)$ in the least multiple of $a$ and $b.$ – hkju Jul 11 '12 at 7:45
Please, write up a definition of $r_i$ and edit it into the body of the question. – Gerry Myerson Jul 11 '12 at 9:07
Note that $R_n | R_m$ if $n|m$. – hkju Jul 11 '12 at 18:13