1. Identify the primes less than 20:
The primes less than 20 are: $2, 3, 5, 7, 11, 13, 17, 19$.

2. **Determine the range of $n$ for each prime $p$**:
For a prime $p$, $n$ must be divisible by $p$ and $p > \sqrt{n}$. This implies $n$ must be in the range $p \leq n < p^2$.

3. **Count the number of valid $n$ for each prime $p$**:
For each prime $p$, the valid $n$ are $p, 2p, 3p, \ldots, (p-1)p$. The number of such $n$ is $p-1$.

4. Sum the counts for all primes:
- For $p = 2$: The valid $n$ are $2$, so there is $1$ number.
- For $p = 3$: The valid $n$ are $3, 6$, so there are $2$ numbers.
- For $p = 5$: The valid $n$ are $5, 10, 15, 20$, so there are $4$ numbers.
- For $p = 7$: The valid $n$ are $7, 14, 21, 28, 35, 42$, so there are $6$ numbers.
- For $p = 11$: The valid $n$ are $11, 22, 33, 44, 55, 66, 77, 88, 99, 110$, so there are $10$ numbers.
- For $p = 13$: The valid $n$ are $13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156$, so there are $12$ numbers.
- For $p = 17$: The valid $n$ are $17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 272$, so there are $16$ numbers.
- For $p = 19$: The valid $n$ are $19, 38, 57, 76, 95, 114, 133, 152, 171, 190, 209, 228, 247, 266, 285, 304, 323, 342$, so there are $18$ numbers.

5. Sum the counts:
$$1 + 2 + 4 + 6 + 10 + 12 + 16 + 18 = 69$$

The final answer is $\boxed{69}$.