There is a certain demand for small and even very small (but not totally random) primes in the

Conjectures 'R Us and

No Prime Left Behind subforums (and others).

For example:

here, or

here, or

here. M.Kamada is also interested in small primes and PRPs: one can take any series (

here for example) and extend it.

In contrast, if one wants to do numbers that are not of interest to anyone else, then the only way to get people excited is to get the reasonably big (to enter top-5000 or, a bit easier target, the PRP Lifchitz site). There needs to be a system that these numbers support

...or else of course one can simply start an endless Pari/GP loop

Code:

p=10^1567;
while(1,p=nextprime(p+1);print(p))

and get a cookie every time it will print something.