\begin{array}{rcl}
P(A) & = & \displaystyle\sum_{i=1}^\infty P(H_i)P(A|H_i) = \\
& = & \displaystyle\sum_{i=1}^\infty 2^{-i}P(Y\in\{i, 2i, 3i, \ldots\}) = \\
& = & \displaystyle\sum_{i=1}^\infty 2^{-i}\sum_{j=1}^\infty 2^{-ij} = \\
& = & \displaystyle\sum_{i=1}^\infty 2^{-i}\frac{2^{-i}}{1-2^{-i}} = \\
& = & \displaystyle\sum_{i=1}^\infty \frac{1}{2^i(2^i-1)} = \\
& = & \displaystyle\sum_{i=1}^\infty\left(\frac{1}{2^i-1}-\frac{1}{2^i}\right)
\end{array}