Vtgyjfy

It's essentially the same as Jyrki Lahtonen's answer, but they invited me, so here's mine. Well, it's the same until the part where I go into detail about estimating where in that interval of potential values we actually get the first pair of equal values.

Let the sequence $a_n$, for $n=1,2,dots$, be defined as $leftlfloor frac xnrightrfloor$ for some positive real $x$. We seek the least $n$ for which $a_n=a_{n+1}$, or equivalently the greatest $a$ which appears twice in the sequence. Also, for convenience, define $b_n=frac xn$. Note that the differences $b_n-b_{n+1}=frac{x}{n(n+1)}$ form a decreasing sequence.

First, a fact about the floor and ceiling function: $lfloor urfloor + lfloor vrfloorle lfloor u+vrfloor le lfloor urfloor + lceil vrceil$, with equality when $v$ is an integer. What does this mean for our sequence $a_n$? When $frac xn - frac x{n+1} ge 1$, $a_n-a_{n+1}ge 1$ as well; we can't have two consecutive entries equal until $b_n$ has two consecutive entries that differ by less than $1$. From that, we will get our lower bound: if $a_n=a_{n+1}$, $b_n-b_{n+1}=frac{x}{n(n+1)}<1$ and $x < n^2+n=(n+frac12)^2-frac14$. Solving for $n$ in terms of $x$, $n > sqrt{x+frac14}-frac12$. Let $$N(x)=leftlfloorsqrt{x+frac14}+frac12rightrfloor$$
be the first integer value that get us the inequality.

Now, for the upper bound. No matter how far we go, until $a_n$ drops all the way to zero, we still have the chance of $a_n$ and $a_{n+1}$ being different. Looking at one difference just won't be enough. Instead, we stack differences together; if $a_n-a_{n+k} < k$, then since $a_n$ is a decreasing sequence of integers, some two consecutive values in that range must be zero. By the other half of our key inequality, this is guaranteed to happen when $b_n-b_{n+k} le k-1$. Start at the first possible place for two values to be equal; we're looking for the least $k$ such that $b_{N(x)}-b_{N(x)+k} le k-1$. This inequality becomes
$$k-1 ge frac{x}{N(x)}-frac{x}{N(x)+k} = frac{kx}{N(x)(N(x)+k)}$$
$$(k-1)N^2(x)+k(k-1)N(x) ge kx$$
This is - well, it's a mess, because of the floor in the definition of $N$. So, we approximate - $N(x) le sqrt{x+frac14}+frac12$, so $x ge (N(x)-frac12)^2-frac14=N^2(x)-N(x)$. Oops - we actually need a lower bound for $x$ here (see comments). If
$$(k-1)N^2(x)+k(k-1)N(x) ge k(N^2(x)+N(x))$$
$$(k^2+2k)N(x) ge N^2(x)$$
$$N(X)le (k+1)^2-1$$
then, since $k(N^2(x)+N(x)) > kx$, $(k-1)N^2(x)+k(k-1)N(x) > kx$ and we have a $k$ that works. This is true precisely when $k>sqrt{N(x)+1}-1$; we will, of course, take the first successful value. The least $n$ with $a_n=a_{n+1}$ must satisfy
$$sqrt{x}approx N(x) le n le N(x)+lfloorsqrt{N(x)+1}rfloor-1 $$
$$le leftlfloorsqrt{x+frac14}+frac12rightrfloor + leftlfloorsqrt{sqrt{x+frac14}+frac32}rightrfloor-1approx sqrt{x}+sqrt[4]{x}$$

And now, for something new.

Where in that interval will it happen? For a randomly chosen $x$, it's essentially random - but biased. The deviation $1-(b_n-b_{n+1})$ increases approximately linearly with $n$ starting at zero for $n=N(x)$, so the sum of $j$ of them grows like $j^2$. The probability of our first duplication coming in the first $j$ chances is thus approximately proportional to $j^2$, and the location follows a wedge distribution; the probability of it being at $N(x)+j$ is approximately $frac{2j+1}{N(x)}$ for $0le j<sqrt{N(x)}$.

But we can do better than that. Write $N^2(x)=x+c$; rearranging our inequalities $N^2-Nle x< N^2+N$,
$$x-N(x) < N^2(x) le x+N(x)$$
Then $frac{x}{N(x)}=frac{N^2(x)+c}{N(x)}=N(x)+frac{c}{N(x)}$. This fractional part $frac{c}{N(x)}$, between $-1$ and $1$, is what actually determines where in the interval we finally reach a spot with two consecutive $a_n$ equal. As we repeatedly subtract quantities slightly less than $1$ from $b_n$, its fractional part increases until it ticks over an integer - and when that happens, we get our first repeat in the $a_n$.

Let $frac{x}{N(x)+k}=N(x)-k+e_k$. As already noted, $e_0=frac{c}{N(x)}in [-1,1)$. For $e_0in [-1,0)$, we seek the first $k$ such that $e_k ge 0$. For $e_0in [0,1)$, we seek the first $k$ such that $e_k ge 1$. We will then have $a_{N(x)+k-1}=a_{N(x)+k}$. Clear the denominator to get
$$x = N^2(x) - k^2 + N(x)e_k + ke_k$$
$$0 = N(x)(e_k-e_0) + ke_k - k^2$$
$$k = frac{e_k +sqrt{k^2+4(e_k-e_0)N(x)}}{2}$$
For negative $e_0$, the key point comes when $e_kapprox 0$, and $2kapprox sqrt{k^2-4e_0 N(x)}$, or $3k^2approx -4e_0 N(x)$ and $kapprox frac{2}{sqrt{3}}sqrt{-e_0 N(x)}$. For positive $e_0$, the key point comes when $e_kapprox 1$, and $2k-1approx sqrt{k^2+4(1-e_0) N(x)}$. Solve that to $kapprox frac{2}{sqrt{3}}sqrt{(1-e_0)N(x)}+frac23$.

So then, the amount $k$ we need to add to $N(x)$ is about $frac{2}{sqrt{3}}sqrt{N(x)}$ times the square root of either $-e_0$ or $1-e_0$. It takes longest when $e_0$ is equal to $-1$ or $0$, at $x=N^2-N$ or $x=N^2$, and shortest when $x$ is slightly less than one of those values. And that's all I have to say on this one.

It cannot occur between term $n$ and term $n+1$ if $frac{x}{n} - frac{x}{n+1} ge 1$, equivalently $x ge n^2 + n$, equivalently $n le -frac{1}{2} + frac{sqrt{1+4x}}{2}$.

It must occur, either between term $n$ and $n+1$, or between term $n+1$ and $n+2$, if $frac{x}{n} - frac{x}{n+1} le frac{1}{2}$, equivalently $x le frac{1}{2} n^2 + frac{1}{2} n$, equivalently $n ge -frac{1}{2} + frac{sqrt{1+8x}}{2}$.

So the first place it appears is somewhere between the two extremes of $-frac{1}{2} + frac{sqrt{1+4x}}{2}$ and $-frac{1}{2} + frac{sqrt{1+8x}}{2} + 1 = frac{1}{2} + frac{sqrt{1+8x}}{2}$.

Proffering the following.

Let $n$ be an integer such that $n^2-nge x$. The smallest such $n$ is $approxsqrt x$.

Let $kge sqrt{n}$ be an integer. Then we have
$$
frac x{n}-frac x{n+k}-(k-1)=frac{kx-(k-1)n(n-k)}{n(n+k)}=frac{k left(-n^2+n+xright)+left(n^2-k^2nright)}{n (k+n)}.
$$
In the last form both expressions in parens in the numerator are negative. Therefore
$$
frac x{n}-frac x{n+k}<k-1.
$$
So while the denominator $m$ increases from $n$ to $n+k$, the quotient $x/m$ decreases by less than $k-1$. This implies that $lfloor x/m rfloor$ can take at most $k$ values when $m$ covers the range $[n,n+k]$, $k+1$ choices, implying that a repetition took place somewhere in that interval.

Lee Mosher already explained why the repetition cannot happen sooner, so this is the first repetition.

The first repeated value of $lfloor x/m rfloor$ occurs somewhere when $m$ is in the interval roughly between $x^{1/2}$ and $x^{1/2}+x^{1/4}$, as first observed by jmerry. See their answer for more details about where within that range we can expect to see a repetition.

$color{brown}{textbf{Preliminary Notes.}}$

$dfrac xN > dfrac x{N+1},$ so the required equality
$$leftlfloordfrac xNrightrfloor = leftlfloordfrac x{N+1}rightrfloor,tag1$$
where $lfloor arfloor = mathrm{floor},(a),$

has solutions iff
$$left{dfrac xNright} > left{dfrac x{N+1}right}.tag2$$
Taking in account that
$$kleft{dfrac xkright} = xhspace{-12mu}mod k,$$
inequality $(2)$ can be presented in the form of
$$dfrac{xhspace{-12mu}mod N}N > dfrac{xhspace{-12mu}mod (N+1)}{N+1},tag3$$
and from $(3)$ should
$$xhspace{-12mu}mod N ge xhspace{-12mu}mod (N+1).tag4$$
Formula $(4)$ can be used for the testing, instead the issue one.

$color{brown}{textbf{Decision.}}$

The least solution $N$ of equality $(1)$ belongs to the interval $xin(n(n-1),n(n+1)],$ so
$$x=n(n-1)+m,quad m=x-(n^2-n),quad min[1,2n],tag5$$
wherein
$$n = leftlceildfrac{sqrt{4x-3}+1}2rightrceiltag6,$$
$lceil arceil = mathrm{ceil},(a).$

Let
$$N=n+k,quad kinmathbb Ntag7,$$
$$dfrac xN = dfrac{n^2-n+m}{n+k}=n-k-1+dfrac{k(k+1)+m}{n+k},$$
$$dfrac x{N+1} = dfrac{n^2-n+m}{n+k+1}=n-k-2+dfrac{(k+1)(k+2)+m}{n+k+1},$$

If $underline{min[1,n-1]}$ then the least solution of $(3)$ can be achieved iff
$$(k+1)(k+2)+mge n+k+1,quad k =leftlceilsqrt{n-m}-1rightrceil,$$
$$N = n - 1 + leftlceilsqrt{n^2-x}rightrceil.$$

If $underline{min[n,2n-1]}$ then the least solution of $(3)$ can be achieved iff
$$(k+1)(k+2)+mge 2(n+k+1),quad k^2+k-(2n-m)geq 0,quad k = leftlceildfrac{sqrt{8n-4m+1}-1}2rightrceil,$$
$$N = n + leftlceildfrac{sqrt{4(n^2+n-x^2)+1}-1}2rightrceil.$$

If $underline{m=2n}$ then
$$dfrac xN = dfrac{n^2+n}{n+k}=n-k+1+dfrac{k(k-1)}{n+k},$$
$$dfrac x{N+1} = dfrac{n^2+n}{n+k+1}=n-k+dfrac{k(k+1)}{n+k+1},$$
and the least solution of $(3)$ can be achieved iff
$$k(k-1)<n+k,quad (k-1)^2<n+1,quad k = leftlceilsqrt{n+1}rightrceil,$$
$$N = n + leftlceilsqrt{n+1}rightrceil.$$

$color{brown}{textbf{Result.}}$

Therefore, the least solution of $(1)$ is
$$boxed{begin{align}
&N=begin{cases}
n - 1 +leftlceilsqrt{n^2-x}rightrceil,quadtext{if}quad x-n(n-1)in[1,n-1]\[4pt]
n + leftlceildfrac{sqrt{4(n^2+n-x)+1}-1}2rightrceil,quadtext{if}quad x-n(n-1)in[n,2n-1]\[4pt]
n + leftlceilsqrt{n+1}rightrceil,quadtext{if}quad x = n(n+1),
end{cases}\
&text{where}\
&n = leftlceildfrac{sqrt{4x-3}+1}2rightrceil.
end{align}}tag8$$

$color{brown}{textbf{Examples.}}$

$underline{x=2475,quad n=50,quad x-n(n-1)=25}.$

There is the first case of $(8).$

Result is $N=54,$ with $leftlfloordfrac{2475}{54}rightrfloor = leftlfloordfrac{2475}{55}rightrfloor=45.$

$underline{x=2500,quad n=50,quad x-n(n-1)=50}.$

There is the second case of $(8).$

Result is $N=57,$ with $leftlfloordfrac{2500}{57}rightrfloor = leftlfloordfrac{2500}{58}rightrfloor=43.$

$underline{x=2450,quad n=49,quad x=n(n+1)}.$

There is the third case of $(8).$

Result is $N=57,$ with $leftlfloordfrac{2450}{57}rightrfloor = leftlfloordfrac{2450}{58}rightrfloor=42.$