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,... is defined thus: an is the smallest positive integer number such that an+an1+...+a1 is positive integer.

Prove that the sequence a1,a2,a3,... contains all positive integers numbers.

My work:

Let a1=1. Then a2+1 is positive integer and a2 is the smallest positive integer then a2=3.

Then a3+2 is positive integer and a3 is the smallest positive integer then a3=2.

Then a4+a3+a2+a1=a4+2 is positive integer and a4 is the smallest positive integer and a4a1,a2,a3 then a4=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

First 1000 terms:
First 1000 terms

First 5000 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.

Attribution
Source : Link , Question Author : Roman83 , Answer Author : Ivan Neretin

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