If you take the sequence of primes, and concatenate the first n of them together, how many of these are prime? It’s not hard to see that 2, 23, and even 2357 is prime. The next term has 355 digits:

235711131719232931374143475359616771737983899710110\
310710911312713113713914915115716316717317918119119\
319719921122322722923323924125125726326927127728128\
329330731131331733133734734935335936737337938338939\
740140941942143143343944344945746146346747948749149\
950350952152354154755756356957157758759359960160761\
3617619631641643647653659661673677683691701709719

The concatenation of primes up until 1033 also appears to be prime.

The On-Line Encyclopedia of Integer Sequences has this sequence as well.

[tags]Primes, Mathematics[/tags]