# Barely an Integer!

Number Theory Level 5

$\large{n = \left \lfloor m \right \rfloor + \left \lfloor \dfrac{m}{2!} \right \rfloor + \left \lfloor \dfrac{m}{3!} \right \rfloor + \ldots + \left \lfloor \dfrac{m}{k!} \right \rfloor + \ldots }$

For every natural number $$n$$, define $$S(n)$$ to be the unique integer $$m$$ (if it exists) which satisfies the equation above. Submit the value of $$S(3438)$$ as your answer.

Bonus: Does there exist a number $$k$$ such that, for any non-negative integer $$n$$, at least one of $$S(n+1), \ S(n+2),\ \ldots \ , \ S(n+k)$$ exists?

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