nt.number theory - a weird sequence with a non-integral term

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...

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