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In a recent answer Max Alekseyev provided two recurrences of the form mentioned in the title which stay integer for a long time. However, they eventually fail.

QUESTION Is there any (added: strictly increasing) sequence of positive integers $c_1,c_2,c_3,\dots$ satisfying the relation $$ c_{n+1}=\frac{c_n(c_n+n+d)}n $$ for for all $n\geq 1$ (and some integer constant $d$)?

(As Sam Hopkins notes, it would be also very interesting if the indices of the sequence start from some $k>1$ rather than from 1.)

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    $\begingroup$ We can express this as a time-invariant dynamical system by adding an additional variable $n$, i.e. $(x,y) \to (x+1, y(y+x+d)/x)$. For each prime $p$, this naturally gives an algebraic dynamical system on $\mathbb Q_p \times \mathbb Z_p$. One wants to know if there is a $d$ such that all these systems stay inside $\mathbb Z_p \times \mathbb Z_p$ forever. Maybe techniques of $p$-adic algebraic dynamics would be helpful here? $\endgroup$
    – Will Sawin
    Commented Sep 27, 2017 at 15:48
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    $\begingroup$ A note that you may want to allow the sequence to start at $c_k$ for some $k>1$ because otherwise the motivating example is not really of this form. $\endgroup$
    – Sam Hopkins
    Commented Sep 27, 2017 at 17:05
  • $\begingroup$ If we ignore the division by n, a necessary condition for the integrality of the original sequence is for the modified $c_{p+1}$ to divide $p$ for all primes $p$. Might it be possible to show that given any $c_1$ and $d$, that there must exist a prime $p$ such that this is not the case? $\endgroup$
    – AxiomaticSystem
    Commented Dec 10, 2018 at 1:59
  • $\begingroup$ @WillSawin: phrased this way, the transformation sort of looks like those found in the theory of cluster algebras, so maybe if there is a positive answer to the question it could be found via the "Laurent phenomenon." $\endgroup$
    – Sam Hopkins
    Commented Dec 10, 2018 at 15:26

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