Define a sequence $(a\_n)\_{n \geq 1}$ by $$na\_n = 2 + \sum\_{i = 1}^{n - 1} a\_i^2.$$ 


(In particular, $a\_1 = 2$.)


How can you show - preferably **without** using a pc! - that not all terms of the sequence are integral?


And which will be the first such term?


Motivation: nothing interesting to say, it's a random problem which I got from someone - I have no reference - and which interested me. Usually one has to prove that all terms are integral :)


Thoughts: nothing interesting. The terms are quickly getting enormous...



a weird sequence with a non-integral term
Sequences like this are sometimes called Somos sequences (and sometimes Gobel sequences) and you can find information about them at Problem E15 in Guy, Unsolved Problems in Number Theory and in the references Guy gives; also I'm sure typing Somos or Gobel into your favorite search engine will turn up something. 


