The paper is written on consecutive integers $1$ through $n$. Then are deleted all numbers ending in $4$ and $9$ and the rest alternating between $-$ and $+$. Finally, an opening parenthesis is added after each character and at the end of the expression the corresponding number of parentheses: $1 - (2 + 3 - (5 + 6 - (7 + 8 - (10 +...))))$.
Find all numbers $n$ such that the value of this expression is $13$.
1. **Define the sequence and the expression:**
   We are given a sequence of consecutive integers from $1$ to $n$. We need to delete all numbers ending in $4$ and $9$, and then alternate the remaining numbers between $-$ and $+$. Finally, we add parentheses as described to form the expression.

2. **Denote the expression:**
   Let $a_n$ be the value of the expression after performing the described operations on the sequence from $1$ to $n$.

3. **Analyze the pattern:**
   We need to find the value of $a_n$ for different forms of $n$. We will consider $n$ in the form of $5k+1$, $5k+2$, $5k+3$, $5k+4$, and $5k+5$.

4. **Case 1: $n = 5k + 1$**
   \[
   a_{5k+1} = (0 + 1 + 5 + 6 + 10 + 11 + 15 + 16 + \cdots + 5k + 5k + 1) - (2 + 3 + 7 + 8 + 12 + 13 + \cdots + 5k - 3 + 5k - 2)
   \]
   Simplifying the sums:
   \[
   = (1 + 10 + 1 + 20 + \cdots + 1 + 10k + 1) - (5 + 10 + 5 + 20 + \cdots + 5 + 10(k-1) + 5)
   \]
   \[
   = (k + 1 + 10 \cdot \frac{k(k+1)}{2}) - (5k + 10 \cdot \frac{k(k-1)}{2})
   \]
   \[
   = (k + 1 + 5k(k+1)) - (5k + 5k(k-1))
   \]
   \[
   = (k + 1 + 5k^2 + 5k) - (5k + 5k^2 - 5k)
   \]
   \[
   = 6k + 1
   \]

5. **Case 2: $n = 5k + 2$**
   \[
   a_{5k+2} = a_{5k+1} - (5k + 2)
   \]
   \[
   = 6k + 1 - 5k - 2
   \]
   \[
   = k - 1
   \]

6. **Case 3: $n = 5k + 3$**
   \[
   a_{5k+3} = a_{5k+2} - (5k + 3)
   \]
   \[
   = k - 1 - 5k - 3
   \]
   \[
   = -4k - 4
   \]

7. **Case 4: $n = 5k + 4$**
   \[
   a_{5k+4} = a_{5k+3}
   \]
   \[
   = -4k - 4
   \]

8. **Case 5: $n = 5k + 5$**
   \[
   a_{5k+5} = a_{5k+4} + (5k + 5)
   \]
   \[
   = -4k - 4 + 5k + 5
   \]
   \[
   = k + 1
   \]

9. **Solve for $a_n = 13$:**
   \[
   13 = 6k + 1 \implies k = 2 \implies n = 5k + 1 = 11
   \]
   \[
   13 = k - 1 \implies k = 14 \implies n = 5k + 2 = 72
   \]
   \[
   13 = k + 1 \implies k = 12 \implies n = 5k + 2 = 62
   \]

The final answer is $\boxed{11, 72, 62}$.