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Divisibility of sum of divisors

Problem 565 Published on Sunday, 19th June 2016, 10:00 am; Solved by 438;
Difficulty rating: 35%

Let $\sigma(n)$ be the sum of the divisors of $n$.
E.g. the divisors of 4 are 1, 2 and 4, so $\sigma(4)=7$.

The numbers $n$ not exceeding 20 such that 7 divides $\sigma(n)$ are: 4,12,13 and 20, the sum of these numbers being 49.

Let $S(n , d)$ be the sum of the numbers $i$ not exceeding $n$ such that $d$ divides $\sigma(i)$.
So $S(20 , 7)=49$.

You are given: $S(10^6,2017)=150850429$ and $S(10^9 , 2017)=249652238344557$.

Find $S(10^{11} , 2017)$