It is known that the Somos-$k$ sequences for $k\ge 8$ do not give integers. But the first terms of **Somos-8** sequence $s_n=a_n/b_n$
$$1, 1, 1, 1, 1, 1, 1, 1, 4, 7, 13, 25, 61, 187, 775, 5827, 14815,\frac{420514}{7}, \frac{28670773}{91}$$
defined by $s_1=s_2=s_3=s_4=s_5=s_6=s_7=s_8=1$,
$$s_{n+8}s_n=s_{n+7}s_{n+1}+s_{n+6}s_{n+2}+s_{n+5}s_{n+3}+s_{n+4}^2\qquad(n\ge1)$$
have only odd denominators $b_n$. Morever $s_n$ has even numerator $a_n$ only for $n=9k$. It was checked for $n\le 67$. First terms of $s_n\mod 8:=a_n\cdot b_n^{-1}\mod 8$ are
$$\begin{array}{l}
1, 1, 1, 1, 1, 1, 1, 1, {\bf 4},\\
7, 5, 1, 5, 3, 7, 3, 7, {\bf 6},\\
7, 7, 5, 7, 5, 3, 1, 1, {\bf 6},\\
5, 1, 5, 5, 3, 3, 7, 1, {\bf 4},\\
7, 7, 3, 3, 7, 7, 3, 1, {\bf 2},\\
1, 3, 1, 3, 7, 5, 3, 5, {\bf 2},\\
5, 7, 3, 7, 7, 3, 7, 3, {\bf 0},\\
1, 3, 7, 3,\ldots\end{array}$$

Is it possible to prove that the sequence $s_n\mod 2$ is periodic?

**EDT.** It was found by მამუკა ჯიბლაძე that numerator of $s_{71}$ is even while that of $s_{72}$ is odd. The last line of the table above is
$$1, 3, 7, 3, 1, 5, 1, {\bf 2}, 5$$
So the conjecture about numerators of $s_n$ is wrong. But the qustion about periodicity is still valid.

denominator... $\endgroup$5more comments