# 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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