In an attempt to copy down from the board a sequence of six positive integers in arithmetic progression, a student wrote down the five numbers, \[ 113,137,149,155,173,  \] accidentally omitting one. He later discovered that he also miscopied one of them. Can you help him and recover the original sequence?
1. **Identify the common difference:**
   The sequence is in arithmetic progression, so the common difference \(d\) between consecutive terms should be constant. Let's denote the sequence as \(a, a+d, a+2d, a+3d, a+4d, a+5d\).

2. **Calculate the differences between the given terms:**
   \[
   137 - 113 = 24, \quad 149 - 137 = 12, \quad 155 - 149 = 6, \quad 173 - 155 = 18
   \]
   These differences are not consistent, indicating that there is a miscopied number and a missing number.

3. **Assume the first term \(113\) is correct and find the common difference \(d\):**
   Since the differences are inconsistent, we need to test possible values for \(d\). Let's start with the smallest difference, \(d = 12\).

4. **Construct the sequence with \(d = 12\):**
   \[
   113, 113+12, 113+2(12), 113+3(12), 113+4(12), 113+5(12)
   \]
   \[
   113, 125, 137, 149, 161, 173
   \]

5. **Check if the sequence fits the given numbers:**
   The given numbers are \(113, 137, 149, 155, 173\). We see that \(155\) does not fit into the sequence \(113, 125, 137, 149, 161, 173\). Therefore, \(155\) must be the miscopied number.

6. **Verify the sequence:**
   The correct sequence should be \(113, 125, 137, 149, 161, 173\). This sequence has a common difference of \(d = 12\) and fits the pattern of an arithmetic progression.

7. **Conclusion:**
   The original sequence, with the correct numbers and the missing number included, is:
   \[
   \boxed{113, 125, 137, 149, 161, 173}
   \]