Quote:
Originally Posted by a1call
There is currently no established way of showing the integer in a reduced form, but it would be quite easy to invent one.

I am not sure if you are aware or not  your sequence is a recursive quadratic polynomial. You can just specify the seed and the depth level.
x_next=f(x) where f(x)=x^2x+1
2>3>7>43>
For sieve:
Factors would be of format factor==1 (mod 6)
Also given the recurrent nature you can easily calculate which depth level a prime p will divide.