Distribution Of Divisor Sums Modulo 32

Number Theory Level 5

For each integer $2 \le k \le n$ , choose a divisor $d_k$ of $k$ , uniformly at random from the set of divisors of $k.$ We denote by $P(n)$ the probability that

$d_2 + d_3 + \cdots + d_n$

is divisible by 32.

Amazingly, there exists a positive integer $N$ such that for all $n\ge N$ , the value of $P(n)$ is exactly $\frac1{32}$ . What is the smallest $N$ for which this is true?