Concatenation problem

[David W. Wilson]

I thought of the following idea for a sequence.

Let

    a(1) = n
    a(n+1) = k where concat(a(i),k) and concat(k,a(i)) are both prime for
all 1 <= i <= k.

In other words, the elements are chosen greedily so that the concatenation
of any two distinct elements is prime. The elements themselves are not
necessarily prime.

If gcd(a(1),10) != 1, then concat(k,a(1)) is never prime, and the sequence
is the singleton sequence (a(1)). For other choices of a(1) I get

    a(1)   sequence
    1      (1,3,7,109,25597,...?)
    3      (3,7,109,673,...?)
    7      (7,9,19,433,...?)
    9      (9,19,391,8491,...?)
    11     (11,23,81,1871,...?)

In each case, it looks empirically as if the sequence is finite. I suspect
there may be some modular argument to that effect, I just can't seem to work
it out.