It is not an answer just a piece of fun.

I've made a mistake in my program and calculated the sequence with wrong recurrence $$a_{n+6}=\frac{a_{n+5}\cdot a_{n+1}+a_{n+4}\cdot a_{n+2}\cdot {a_{n+3}}^2}{a_n}\ \ (n\ge0).$$ In both cases (1) $a_0=a_1=a_2=a_3=a_4=a_5=1$ and (2) $a_0=a_1=a_2=a_3=1$, $a_4=a_5=2$ it gives at least $40$ integer values: $$\tag{1}1, 1, 1, 1, 1, 1, 2, 3, 5, 17, 107, 1489, 79541, 96735414,\ldots$$ $$\tag{2}1, 1, 1, 1, 2, 2, 4, 12, 44, 472, 13144, 5509040, 32227528976,\ldots$$

It is not an answer just a piece of fun.

I've made a mistake in my program and calculated the sequence with wrong recurrence $$a_{n+6}=\frac{a_{n+5}\cdot a_{n+1}+a_{n+4}\cdot a_{n+2}\cdot {a_{n+3}}^2}{a_n}\ \ (n\ge0).$$ In both cases (1) $a_0=a_1=a_2=a_3=a_4=a_5=1$ and (2) $a_0=a_1=a_2=a_3=1$, $a_4=a_5=2$ it gives at least $40$ integer values: $$\tag{1}1, 1, 1, 1, 1, 1, 2, 3, 5, 17, 107, 1489, 79541, 96735414,\ldots$$ $$\tag{2}1, 1, 1, 1, 2, 2, 4, 12, 44, 472, 13144, 5509040, 32227528976,\ldots$$

EDT: These sequences are really integers. It can be proved with simple (George Bergman's) argument described in David Gale's The strange and surprising saga of the Somos sequences, see Tracking the automatic ant and other mathematical explorations.