# nt.number theory – Composite sequence possibly only integers

I will first ask the question that can be asked very simply. Then I will explain a motivation and give references to related sequences.

Consider the sequence defined by
$$b_n = frac {(b_ {n-1} +1) (b_ {n-2} + 1) (b_ {n-3} + 1) (b_ {n-4} +1)} {b_ {n-5}}$$
or $$b_0 = b_1 = b_2 = b_3 = b_4 = 1$$.

Question: is $$b_n in mathbb {Z}$$ for everyone $$n geq 0$$?

Provided that my program is correct, I found that $$b_n in mathbb {Z}$$ for $$0 leq n leq 36$$.

I arrived at this reflection on MO323867 which requires sequences which have all values ​​but do not satisfy the Laurent phenomenon.
More generally, we can consider the sequence defined by

$$a ^ {(k)} _ n = frac { prod_ {i = 1} ^ {k-1} (a ^ {(k)} _ {ni} + 1)} {a ^ {(k )} _ {nk}}$$

with $$a_0 = a_1 = cdots = a_ {k-1} = 1$$.

• When $$k = 2$$ we obtain $$a ^ {(2)} _ n a ^ {(2)} _ {n-2} = a ^ {(2)} _ {n-1} + 1$$ which can be seen coming from a guy $$A_2$$ Cluster algebra that implies the Lawrence phenomenon and completeness (see A076839).

• When $$k = 3$$ we have the Laurent phenomenon and consequently the sequence is entirely (see A276123). The Laurent phenomenon follows for a more general result in Theorem 6.2.3 of Matthew Russell's thesis.

• When $$k = 4$$ the sequence does not present the Laurent phenomenon, but it is always an entire sequence (see A276175, MO248604, MSE1905063).

• When $$k = 5$$ we have the sequence of the question posed above.

• When $$k> 5$$ the sequence does not consist solely of integers, the first violation being $$a ^ {(k)} _ {2k} not in mathbb {Z}$$.
This can be seen by looking at the 2-adic valuation. In particular, we can show $$nu_2 (a ^ {(k)} _ {2k}) = 5-k$$ for $$k geq 5$$. To see this, we can work $$pmod {2 ^ 5}$$ and finds that $$(a ^ {(k)} _ {2k-1} + 1) equiv 2 ^ 4 pmod {2 ^ 5}$$. It's clear that $$a_k = 2 ^ {k-1}$$ and $$a_ {n}$$ is even, equivalently $$(a_ {n} +1)$$ is strange, for $$k leq n <2k – 1$$.

So the sequence $$b_n$$ correspond to $$k = 5$$ is the only sequence of the family for which I do not know if all the terms are integers.