# ana_n is the smallest positive integer number such that √an+√an−1+…+√a1\sqrt{a_n+\sqrt{a_{n-1}+…+\sqrt{a_1}}} is positive integer

An infinite sequence of pairwise distinct numbers $$a1,a2,a3,...a_1, a_2, a_3, ...$$ is defined thus: $$ana_n$$ is the smallest positive integer number such that $$√an+√an−1+...+√a1\sqrt{a_n+\sqrt{a_{n-1}+...+\sqrt{a_1}}}$$ is positive integer.

Prove that the sequence $$a1,a2,a3,... a_1, a_2, a_3, ...$$ contains all positive integers numbers.

### My work:

Let $$a1=1a_1=1$$. Then $$√a2+1\sqrt{a_2+1}$$ is positive integer and $$a2a_2$$ is the smallest positive integer then $$a2=3a_2=3$$.

Then $$√a3+2\sqrt{a_3+2}$$ is positive integer and $$a3a_3$$ is the smallest positive integer then $$a3=2a_3=2$$.

Then $$√a4+√a3+√a2+√a1=√a4+2\sqrt{a_4+\sqrt{a_3+\sqrt{a_{2}+\sqrt{a_1}}}}=\sqrt{a_4+2}$$ is positive integer and $$a4a_4$$ is the smallest positive integer and $$a4≠a1,a2,a3a_4\not=a_1,a_2,a_3$$ then $$a4=7a_4=7$$.

(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 1000 terms:

First 5000 terms:

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.