First Harmonic Number to exceed 20

Theorem
The first harmonic number that is greater than $20$ is $H_{272 \, 400 \, 600}$.

That is, the number of terms of the harmonic series required for its partial sum to exceed $20$ is $272 \, 400 \, 600$.

Proof
We have:
 * $H_{272 \, 400 \, 599} = \ds \sum_{k \mathop = 1}^{272 \, 400 \, 599} \frac 1 k \approx 19 \cdotp 99999 \, 99979$

and:
 * $H_{272 \, 400 \, 600} = \ds \sum_{k \mathop = 1}^{272 \, 400 \, 600} \frac 1 k \approx 20 \cdotp 00000 \, 00016$