[seqfan] Very bounded sequences

[Tomasz Ordowski]

Dear SeqFans,

I defined an interesting sequence:

a(0) = 3; a(n) = smallest k > 1 such that 1 + a(0)a(1)...a(n-1)k is
composite.

3, 3, 3, 2, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2,
3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, ...

a(n) = 4 for n = 4 and 39,
a(n) = 3 for n = 0, 1, 2, 5, 6, 7, 14, 20, 25, 56, 90, 119, 316, 330, 1268,
1604, 1805, 1880, 1984, 2950, 3386, 3712, 4532, 4874, 8968, 18178, 19454,
...
a(n) = 2 for others n < 20000.
Data from Amiram Eldar.

Similar tails have sequences with other initial terms that are natural
numbers.

Conjecture: For any initial term a(0) > 0, a(n) > 3 only for finitely many
n >= 0.

The question is how to prove that all these sequences are bounded, so
bounded?

It seems that a(0) = 21 is the smallest initial term such that a(n) = 2 or
3 for every n > 0.

Note that if a(0) is a Sierpinski number, then a(n) = 2 for every n > 0.

How to explain the occurrence of only twos and threes in the tails?

Are similar sequences described in the literature?

Best regards,

Thomas Ordowski