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.
Not an answer but some additional graphics to @Ivan Neterin's answer (not an answer).

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I was wondered not about $a\_n$'s but those integers $$ M\_n = \sqrt{a\_n + \sqrt{a\_{n-1} + ...}}$$
And what I find? BOOM. Unexpected (at least for me).
[![enter image description here](https://i.stack.imgur.com/Hlcyi.png)](https://i.stack.imgur.com/Hlcyi.png)
I do not know if it is some visual noise or not but you can notice kind of parabolic curves on pictures. Interesting what is their nature.
[![enter image description here](https://i.stack.imgur.com/W0Czb.png)](https://i.stack.imgur.com/W0Czb.png)