From $p - q = q - r$, we get $2q = p + r$, so $p$, $q$, $r$ are in an arithmetic progression (AP). All valid triples must use odd primes (2 cannot fit in a 3-term prime AP under 20).

Primes < 20: 2, 3, 5, 7, 11, 13, 17, 19

All 3-term APs of primes under 20:

• (3, 5, 7) → sum = 15
• (3, 7, 11) → sum = 21
• (5, 11, 17) → sum = 33
• (3, 11, 19) → sum = 33
• (7, 13, 19) → sum = 39

Distinct sums: {15, 21, 33, 39} → 4 values.

Hence, the answer is 4.