I have something that might be suitable as a puzzle for your site.

{41,271,311,541,571,661,1061,1181,1361,1931,...}

1) find the next n primes

2) find a closed formula describing this sequence

3) Is it infinite?

4) Is it dense?

I can generate as many terms as I need, but at a high computational cost.
I don't know of a closed formula, but I would like to find one...