Divisibility Streaks

For every positive number $n$ we define the function $s t r e a k (n) = k$ as the smallest positive integer $k$ such that $n + k$ is not divisible by $k + 1$ .
E.g:
$13$ is divisible by $1$
$14$ is divisible by $2$
$15$ is divisible by $3$
$16$ is divisible by $4$
$17$ is NOT divisible by $5$
So $s t r e a k (13) = 4$ .
Similarly:
$120$ is divisible by $1$
$121$ is NOT divisible by $2$
So $s t r e a k (120) = 1$ .

Define $P (s, N)$ to be the number of integers $n$ , $1 < n < N$ , for which $s t r e a k (n) = s$ .
So $P (3, 14) = 1$ and $P (6, 10^{6}) = 14286$ .

Find the sum, as $i$ ranges from $1$ to $31$ , of $P (i, 4^{i})$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.