# Perfect triangular numbers

If we pick any even perfect number less than $$9^{9^{9^{9^{9^9}}}}$$ at random, what is the probability that the perfect number chosen is also a triangular number?

Express the probability as a rational number $$\dfrac ab$$, where $$a$$ and $$b$$ are coprime positive integers. And submit $$\mu\left(a+b\right)\cdot\left(ab\right)^2$$ as your answer.

Notation: $$\mu$$ denotes the Möbius function.

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