Distribution Of Divisor Sums Modulo 32

For each integer 2kn2 \le k \le n, choose a divisor dkd_k of kk, uniformly at random from the set of divisors of k.k. We denote by P(n)P(n) the probability that

d2+d3++dnd_2 + d_3 + \cdots + d_n

is divisible by 32.

Amazingly, there exists a positive integer NN such that for all nNn\ge N, the value of P(n)P(n) is exactly 132\frac1{32}. What is the smallest NN for which this is true?

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