Sequence
 

Generate a sequence where the N+1 term is the product of all previous terms +1 and the t(N+1) is divisible byp(N+1) ie the Nth prime.

SO t(1) divisible by 1
t(2)=t(1)+1 and divisble by 2
t(3)=t(2)*t(1)+1 and divisble by 3
t(4)=t(3)*t(2)*t(1)+1 and divisble by 5.

and so on...

Good luck this is a hard problem, lets see who can generate the longest sequence.

Can you find a algorithm to generate this sequence to some prime p?
Citrix

There are multiple solution till each p but the smallest t(1) will be the correct one.

Citrix

[spoiler]Let T = t(1). Then:
t(2) = T + 1
t(3) = t(2)*t(1) + 1 = (T + 1)*T + 1 = T^2 + T + 1
t(4) = t(3)*t(2)*t(1) + 1 = (T^2 + T + 1)*(T + 1)*T + 1 = T^4 + 2T^3 + 2T^2 + T + 1

By looking modulo 5, the condition that t(4) is divisible by 5 is impossible.
The best sequence is thus:
t(1) = 1
t(2) = 2
t(3) = 3
[/spoiler]

cool solution, but the formula you generated for t4 is prime for 2,4,16,256.

All 2^2^n. Pretty cool. Are there any other primes of the form 2^2^n?

Citrix

Also prime for 2^(2^6)

no primes upto n=18