[seqfan] Primorial Polynomial ?

[David Sycamore]

For any integer n>=1 and any prime q, define:

P_n = prime^n(q) + prime^(n-1)(q) + 
...... + prime(q) + q + (1-(-1)^n)/2.

Where prime^2(q) = prime(prime(q)), etc. 

Then for any n>=1, and any q>2, P_n is always odd, and can sometimes be prime.

Examples:
P^1(5) =  11 + 5 + 1 = 17
P^2(3) = 11 + 5+ 3= 19
P^3(7) = 277 + 59 + 17 + 7 +1 = 361

Let a(n) be the smallest prime q such that P_n(q) is prime. 

(Note that a(n)>2 for all n because P_n(2) is always composite).

According to my calculation, the first few terms are: 5,3,13,5,3,29,3

Would anyone be interested to find more terms?  (a code would be nice). 
Any comments welcome.
Cheers
David.