P with integer factorization oracle I just read the "Is integer factorization an NP-complete problem?" question ... so I decided to spend some of my reputation :-) asking another question $Q$ having $P(\text{Q is trivial}) \approx 1$:

If $A$ is an oracle that solves integer factorization, what is the power of $P^A$?


I think it makes RSA-based public-key cryptography insecure ... but apart from this, are there other remarkable results?
Obviously any decision problem that can be reduced to factoring can be solved with a factoring oracle. But since we're given the ability to make multiple queries, I tried to think of a non-trivial problem for which one would want to make multiple queries.  

The problem of computing the Euler totient function seems like such a problem.  I don't know  how to solve the decision version of this problem by a Karp-reduction to the decision version of factoring.  But with Turing reductions, it's easy to reduce this to factoring.