We have so much in common

Let \(S\) be the set of all positive integers \(n\) such that each of \(n\) and \(n + 1\) have exactly four divisors and have their divisors add to the same value.

Let \(m\) be the number of elements in \(S\) and \(b\) be the sum of these \(m\) elements.

Find \(m + b.\)


Inspired by Paul Ryan Longhas

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