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IlyaShpitser comments on From the "weird math questions" department... - Less Wrong
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If the oracle is fake, you will find out in finite time since eventually you will print N 1s, and the oracle will lie about that program. If the oracle is true, this method will never terminate. But I don't think there is a way for a Turing machine to check for a true oracle (only for a fake oracle!).
What if the finite time needed to determine N is greater than the finite time that you have available?
OP asked "is there a way" not "is there a fast way".
"In finite time" is not the same as "Using a halting process"