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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