Question: An infinite sequence of pairwise distinct numbers $a\_1, a\_2, a\_3, ...$ is defined thus: $a\_n$ is the smallest positive integer number such that $\sqrt{a\_n+\sqrt{a\_{n-1}+...+\sqrt{a\_1}}}$ is positive integer.

Prove that the sequence $ a\_1, a\_2, a\_3, ... $ contains all positive integers numbers.

### My work:

Let $a\_1=1$. Then $\sqrt{a\_2+1}$ is positive integer and $a\_2$ is the smallest positive integer then $a\_2=3$.

Then $\sqrt{a\_3+2}$ is positive integer and $a\_3$ is the smallest positive integer then $a\_3=2$.

Then $\sqrt{a\_4+\sqrt{a\_3+\sqrt{a\_{2}+\sqrt{a\_1}}}}=\sqrt{a\_4+2}$ is positive integer and $a\_4$ is the smallest positive integer and $a\_4\not=a\_1,a\_2,a\_3$ then $a\_4=7$.

Answer: 
*(not an answer, only meant to share my astonishment)*

Look at it, guys! Just freaking **look** at it!

First 50 terms:
1, 3, 2, 7, 6, 13, 5, 22, 4, 33, 10, 12, 21, 11, 32, 19, 20, 31, 30, 43, 9, 45, 18, 44, 29, 58, 8, 60, 17, 59, 28, 75, 16, 76, 27, 94, 15, 95, 26, 115, 14, 116, 25, 138, 24, 163, 23, 190, 35, 42.

First 200 terms:
[![First 200 terms](https://i.stack.imgur.com/VC6nW.png)](https://i.stack.imgur.com/VC6nW.png)

First 1000 terms:
[![First 1000 terms](https://i.stack.imgur.com/flLwZ.png)](https://i.stack.imgur.com/flLwZ.png)

First 5000 terms:
[![First 5000 terms](https://i.stack.imgur.com/6tBnD.png)](https://i.stack.imgur.com/6tBnD.png)

My first guess was that the thing is chaotic, and we won't ever be able to prove a thing. Now I've changed my mind.
It seems to me, from looking at the graphs in Ivan Neretin's answer, that one change of phase is indicated by $a\_i=a\_{i-1}-1$. The change of phase the other way could be defined by the highest $a\_i$ preceding a phase-change of the first kind. I realise that there are alternative definitions of when exactly the phase changes. But with the above definitions, the changes of the first kind are at $i=3, 5, 19, 112, 630, 3637, 21101, 122766, 714992, 4165987, 24278023,\dots$, with $a\_i=2, 6, 30, 182, 1056, 6162, 35910, 209306, 1219920, 7110222, 41441406,\dots$. In each case, the preceding high points were at $i=2, 4, 10, 48, 269, 1521, 8780, 50942, 296383, 1726143, 10057582,\dots$ with $a\_i=3, 7, 33, 190, 1075, 6208, 36021, 209574, 1220567, 7111784, 41445177,\dots$. With each of these four sequences, the ratio of term to preceding term seems to converge to the value $3+2\sqrt2\sim5.82842712474619$ which is the limit of the term-ratio of the recurrence $b\_n=6b\_{n-1}-b\_{n-2}$. (The term-ratios of the above sequence $a\_i=2, 6,\dots$ converge somewhat more quickly than those of the other sequences above.)