An interesting series

[Citrix]

I have been playing with this series, the numbers in the series have been very easy to factor.
Here is how to generate the series.

Let t(n)!= t(n)*t(n-1)*t(n-2)...t(0)

Start with seed s=t(0)

t(1)=smallest factor of (s^2+1) that is already not in the series
t(2)=smallest factor of ((s+t(1)!)^2+1) that is already not in the series
t(3)=smallest factor of ((s+t(1)!+t(2)!)^2+1) that is already not in the series
and so on.

Now the question is that for say seed s=1, do you ever run out of the new terms to add in the series.
Can it be proven that you never run out of factors for a random s?

Any solutions?


A similar problem!
(Series 2)

Start with seed s=t(0)

t(1)=smallest factor of (s^2+1) that is already not in the series
t(2)=smallest factor of ((s+t(1))^2+1) that is already not in the series
t(3)=smallest factor of ((s+t(1)+t(2))^2+1) that is already not in the series
and so on.

For what seeds will you run out of factors and for which you never run out of factors. For seed=1,2 you do.


Citrix