For any positive integer $[n]$, let $[n]=\{1,\ldots,n\}$. Let $r\in\mathbb{R}$. We define for every positive integer $n\in\mathbb{N}$ the *minimal difference from a rational with denominator $\leq n$ to* $r$ by $$\text{md}\_n(r)=\min\{|r-\frac{a}{b}|: a\in\mathbb{Z}, b\in[n] \},$$ and let $$d\_n(r) = \min\{b\in[n]: \exists a\in\mathbb{Z}\big(|r-\frac{a}{b}| = \text{md}\_n(r)\big)\}.$$


So, for every $r\in\mathbb{R}$ we get an increasing sequence of positive integers $d(r) = (d\_n(r))\_{n\in\mathbb{N}}$, which we call the *denominator approximation sequence*. (Does this concept have a proper name?)


For instance we have $d\_8(\pi)=d\_9(\pi)=d\_{10}(\pi) = 7$, as $\frac{22}{7}$ is the best rational approximation to $\pi$ with denominator $\leq 10$. Note that $r\in\mathbb{Q}$ if and only if $d(r)$ is eventually constant.


**Question.** Given an integer sequence $a$ with $a(n)\in[n]$ for all $n\in\mathbb{N}$, is there $r\in\mathbb{R}$ with $a = d(r)$?



Denominator approximation sequence of a real number
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](https://www.isinj.com/mt-usamo/Elementary%20Number%20Theory%20-%20A%20Problem%20Solving%20Approach%20-%20Roberts%20(MIT,%201977).pdf), cited above, is beautiful but hard to find in print.)




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




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




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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 $27,38,49,60$ 




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




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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.




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




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




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


