To solve the problem, we need to find the smallest integer $n$ such that $S(n) - S(5n) = 2013$. Here, $S(k)$ denotes the sum of the digits of $k$.

1. Understanding the Sum of Digits Function:
- For any integer $k$, $S(k)$ is the sum of its digits.
- For example, $S(21) = 2 + 1 = 3$ and $S(105) = 1 + 0 + 5 = 6$.

2. Analyzing the Problem:
- We need to find $n$ such that $S(n) - S(5n) = 2013$.
- This implies that the sum of the digits of $5n$ is significantly smaller than the sum of the digits of $n$.

3. Properties of Digit Sums:
- When multiplying a number by 5, the sum of the digits can change significantly due to carries in the decimal system.
- For example, if $n = 199$, then $5n = 995$, and $S(199) = 1 + 9 + 9 = 19$ while $S(995) = 9 + 9 + 5 = 23$.

4. **Finding the Smallest $n$:**
- We need to find $n$ such that $S(n) - S(5n) = 2013$.
- Let's consider $n$ to be a number with many 9's, as this will maximize $S(n)$ and minimize $S(5n)$.

5. **Constructing $n$:**
- Suppose $n$ has $k$ digits, all of which are 9. Then $n = 999\ldots9$ (with $k$ digits).
- For such $n$, $S(n) = 9k$.
- When $n$ is multiplied by 5, $5n = 4999\ldots95$ (with $k-1$ digits of 9 and the last digit being 5).
- For $5n$, $S(5n) = 4 + 9(k-1) + 5 = 9k$.

6. Calculating the Difference:
- We need $S(n) - S(5n) = 2013$.
- Substituting the values, we get $9k - (4 + 9(k-1) + 5) = 2013$.
- Simplifying, $9k - (9k - 9 + 9) = 2013$.
- This simplifies to $9k - 9k + 9 - 9 = 2013$, which is not possible.

7. Revising the Approach:
- We need to consider the carries more carefully.
- If $n$ has $k$ digits, then $5n$ will have at most $k+1$ digits.
- The sum of the digits of $5n$ will be less than or equal to $9k + 5$.

8. **Finding the Correct $k$:**
- We need $9k - (9k + 5) = 2013$.
- This simplifies to $-5 = 2013$, which is not possible.

9. Correcting the Calculation:
- We need to find $k$ such that $9k - S(5n) = 2013$.
- Let $S(5n) = 9k - 2013$.
- Since $S(5n)$ must be a valid sum of digits, we need to find $k$ such that $9k - 2013$ is a valid sum of digits.

10. **Finding the Smallest $k$:**
- We need $9k - 2013$ to be a valid sum of digits.
- The smallest $k$ that satisfies this is $k = 224$.

Conclusion:
The number of digits in $n$ is $\boxed{224}$.