The prime numbers (1, 2, 3, 5, 7, 11, 13) have only 2 positive integer factors, that is themselves and 1, which leaves us with only non-prime numbers to consider.

The positive integer factors of:

4 are 1, 2, 4 = 3

6 are 1, 2, 3, 6 = 4

8 are 1, 2, 4, 8 = 4

9 are 1, 3, 9 = 3

10 are 1, 2, 5, 10 = 4

12 are 1, 2, 3, 4, 6, 12 = 6

14 are 1, 2, 7, 14 = 4

15 are 1, 3, 5, 15 = 4

16 are 1, 2, 4, 8, 16 = 5

Therefore on 3 integers (4 and 9) between 1 and 16 (inclusive) have three positive integer factors