There are $N$ buckets.
Each second we add one new ball to a random bucket - so at $t=k$, there are a total of $k$ balls collectively in the buckets.
At $t=1$, we expect that at least one bucket contains one ball.
At $t=\sqrt{2N\ln{2}}$, due to birthday paradox, we expect that at least one bucket contains two balls.
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At $t=f(m)$, we expect at least one bucket to contain $m$ balls.
What is the function $f(m)$?