Sets of Numbers!

Algebra Level 5

Let \(S\) be the set of all numbers of the form \(a(n) = n^2 + n + 1\), where \(n\) is a natural number. Find the sum of all positive integral values of \(k\) such that the product \(a(n) \cdot a(n+k) \) also belongs to \(S\) for all natural numbers \(n\).

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