There are Bitcoin-like network that performs calculations that are not locally useful, but interesting for peers.

There are {SETI,folding,...}@home networks that calculates something that is (probabilistically) useful locally, but not such interesting for peers.

Can there be a network that:

  1. Allows starting useful distributed calculations and
  2. Have notions of proof of works, signatures, checking and other building blocks that can be used to run BitCoin-like currency?

Ideally, if somebody finds an efficient way to solve some useful task (compare: start using GPU while everybody else using only CPU), he not just start having larger share in a zero-sum game, but also automatically help that calculational task.