Fulsorials

[a1call]

Won't be anytime soon. I am only hoping it will be in my lifetime.

I have to retool my setup for the next iteration. I intend to try a better sieving method so it will be some time before it's ready.

But thanks for the compliment.

[Citrix]

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^2-x+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.

[a1call]

Originally Posted by Citrix

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^2-x+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.

This is a very old thread and the concept has evolved since the OP.
Your definition seems to relate to N-1 flavour with k=1.
The oeis sequence is the N+1 flavour. There are two primary iteration-flavours and infinite combinations of the 2 are possible. The k-always-equal-1 is problematic since any (large) non-prime iteration will render the later iterations non-provable. The modular logic you point out is very helpful and should speed things up. Thank you very much.
I am lost in your last sentence, but I assume regardless that, the necessary depth will be beyond what can be executed for a 400k dd integer so perhaps we can leave it at that.
Again thanks for the insight.