5. (10 points) Among all positive integers, those whose sum of divisors does not exceed 30 are counted as follows.

保留源文本的换行和格式，这里应该是：

5. (10 points) Among all positive integers, those whose sum of divisors does not exceed 30 are counted as follows.
19
【Answer】Solution: According to the analysis, this positive integer does not exceed 30, so all prime numbers not exceeding 30 meet the condition, which are 2, 
$3, 5, 7, 11, 13, 17, 19, 23, 29$, a total of 10;
Other non-prime numbers are: $1, 4, 6, 8, 9, 10, 12, 14, 15$ for a total of 9 that meet the condition, so the positive integers that satisfy the sum of divisors not exceeding 30 are: $10+9=19$ in total.

Therefore, the answer is: 19.