Sequence Inspired By Sylvester's Sequence

[Leroy Quet]

I was inspired by Sylvester's sequence (sequence A000058: 
a(n)=a(n-1)*(a(n-1)-1)+1) to come up with this recursive sequence where 
every term is coprime with every other:
 
a(1) and a(2) = integers where GCD(a(1),a(2)) = 1.

a(n+2) = a(n) *(a(n) +a(n+1)) - a(n+1).


We can rewrite the recursion as

a(n+2) = - a(n+1) + (a(1)+a(2)) * product{k=1 to n} a(k) 

 From this last recursion we can see that each term is coprime to every 
other.


With a(1) = 2 and a(2) = 3 the sequence begins
2, 3, 7, 23, 187, 4643,...

(Not in EIS. I will submit the terms above myself.)

Is there a direct way of calculating the terms?

What else can be said about this sequence, such as the fraction of terms 
among the first N which are prime?



We can also consider sequences of the form:
a(1) and a(2) are integers where GCD(a(1),a(2))=1.
a(n+2) = a(n)*(a(n+1) + a(n)^m(n)) - a(n+1)^m(n+1) 
or
a(n+2) = a(n)*(a(n+1) - a(n)^m(n)) + a(n+1)^m(n+1),

where {m(k)} is a sequence of positive integers.

Since these recursions are the same as

a(n+2) = +- a(n+1)^m(n+1) + (a(2) -+ a(1)^m(1)) *product{k=1 to n} a(k),

every term is coprime to every other.

I know this is not very advanced stuff, but it is what my math interest 
is at the moment.

thanks,
Leroy Quet