## Dissonant Numbers

### Problem 515

Let `d`(`p`,`n`,0) be the multiplicative inverse of `n` modulo prime `p`, defined as `n` × `d`(`p`,`n`,0) = 1 mod `p`.

Let `d`(`p`,`n`,`k`) = $\sum_{i=1}^n$`d`(`p`,`i`,`k`−1) for `k` ≥ 1.

Let `D`(`a`,`b`,`k`) = $\sum$(`d`(`p`,`p`-1,`k`) mod `p`) for all primes `a` ≤ `p` < `a` + `b`.

You are given:

`D`(101,1,10) = 45`D`(10^{3},10^{2},10^{2}) = 8334`D`(10^{6},10^{3},10^{3}) = 38162302

Find `D`(10^{9},10^{5},10^{5}).