The numbers which are written as the sum of two prime numbers are:

4 = 2 + 2

5 = 2 + 3

6 = 3 + 3

7 = 2 + 5

8 = 3 + 5

9 = 7 + 2

10 = 5 + 5

12 = 7 + 5

13 = 11 + 2

14 = 7 + 7

15 = 13 + 2

16 = 13 + 3

18 = 11 + 7

19 = 17 + 2

So the numbers left are: 1, 2, 3, 11, 17

The sum of all positive integers less than 20 that cannot be written as the sum of two prime numbers is:

1 + 2 + 3 + 11 + 17 = 34