Timeline for Is there any positive integer sequence $c_{n+1}=\frac{c_n(c_n+n+d)}n$?

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Dec 10, 2018 at 15:26	comment	added	Sam Hopkins		@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."
Dec 10, 2018 at 1:59	comment	added	AxiomaticSystem		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?
Sep 27, 2017 at 19:11	history	edited	Ilya Bogdanov	CC BY-SA 3.0	`increasing' added
Sep 27, 2017 at 17:05	comment	added	Sam Hopkins		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.
Sep 27, 2017 at 15:57	history	edited	Max Alekseyev	CC BY-SA 3.0	spelling:)
Sep 27, 2017 at 15:48	comment	added	Will Sawin		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?
Sep 27, 2017 at 15:35	history	asked	Ilya Bogdanov	CC BY-SA 3.0