## Prime triples and geometric sequences

### Problem 518

Let S(`n`) = Σ `a`+`b`+`c` over all triples (`a`,`b`,`c`) such that:

`a`,`b`, and`c`are prime numbers.`a`<`b`<`c`<`n`.`a`+1,`b`+1, and`c`+1 form a**geometric sequence**.

For example, S(100) = 1035 with the following triples:

(2, 5, 11), (2, 11, 47), (5, 11, 23), (5, 17, 53), (7, 11, 17), (7, 23, 71), (11, 23, 47), (17, 23, 31), (17, 41, 97), (31, 47, 71), (71, 83, 97)

Find S(10^{8}).