Problem Archive

For every positive number $n$ we define the function $\mathop{streak}(n)=k$ 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 $\mathop{streak}(13) = 4$.
Similarly:
$120$ is divisible by $1$
$121$ is NOT divisible by $2$
So $\mathop{streak}(120) = 1$.

Define $P(s, N)$ to be the number of integers $n$, $1 \lt n \lt N$, for which $\mathop{streak}(n) = s$.
So $P(3, 14) = 1$ and $P(6, 10^6) = 14286$.

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