Look up rational approximation in Wikipedia to find answers to your questions. It is pretty lovely.

This question does not seem well thought through. Why not just list $1,3,4,9,13$ instead of $1,1,3,4,4,4,4,4,9,9,9,9,13?$

You need only consider $0 \leq r \leq 1/2$ since the sequences for $r$ and $1-r$ have the same denominators.

It seems odd to list only the denominators of the rational approximations rather than the approximations themselves.

**Incorrect statement:** If a certain rational $\frac{a}{b}$ appears in the sequence for $r$ then the sequence for $\frac{a}{b}$ itself is the same sequence truncated at $\frac{a}{b}.$

The correct description is more complicated and depends on considering which continued fractions can have one of $\frac{a}{b}$ as a semi-convergent.

I will suffice with describing the sequences including a $60.$ Some of the beauty of the result can be observed. For details read the article above or a number theory textbook (Roberts, cited above, is beautiful but hard to find in print.)

Near $\frac{1}{60}$ it is $\mathbf{1,s,s+1,\cdots,60,\cdots} \ \ $ for some $30 \leq s \leq 60.$ One can take $r=\frac1{2s}.$

Near $\frac{7}{60}$ it is $\mathbf{1,5,6,7,8,9,*}$ with $*$ first $17,60$ then $17,26,60$ then $17,26,43,60$

near $\frac{11}{60}$ it is $\mathbf {1,3,4,5,6,11,*}$ with $*$ first $60$ then $49,60$ then $38,49,60$

and then $\mathbf{1,3,4,5,11,*}$ with $*$ first $38,49,60$ then $27,38,49,60$

Near $\frac{13}{60}$ it is $\mathbf{1,3,4,5,9,14,23,*}$ with $*$ first $37,60$ then $60$

Near $\frac{17}{60}$ it is $\mathbf{1,2,3,4,7,*}$ with $*$ the last $5,4,3,2$ or $1$ of $32,39,46,53,60$ in that order.

Near $\frac{19}{60}$ it is $\mathbf{1,2,3,*,60}$ with $*$ $10,13,16,19$ or $10,13,16,19,41$ or $13,16,19,41$

Near $\frac{23}{60}$ it is $\mathbf{1,2,3,5,8,13,*}$ with $*$ first $34,47,60$ then $47,60$ then $60.$

Finally, near $\frac{29}{60}$ it is $\mathbf{1,2,15,17,19,21,23,*,60}$ with $*$ first $25,27,29$ then $25,27,29,31$

and then it is $\mathbf{1,2,17,19,21,23,25,27,29,31,60}$