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 $a_4=7$.
Best 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 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.