# 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...

• Another description: $a_1=2$ and for $n>1$, $a_n=a_{n-1}+\frac{a_{n-1}(a_{n-1}-1)}{n}$ – Jonas Meyer Feb 1 '10 at 0:11
• Why a downvote? – Wanderer Feb 3 '10 at 15:37

• I deleted my answer (it is $k=43$, which I found by fairly routine computation) because of this excellent reference. – moonface Feb 1 '10 at 0:25
My 2 cents (ha ha): perhaps the approach is to show that the power of 2 dividing $2+\sum_{i=1}^{n-1}a_i^2$ is eventually less than the power of 2 in $n$, and that this somehow involves looking at $a_{2^n}$. I'll keep playing with it for a bit.