- #1

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([itex]m[/itex] may be a function of [itex]k[/itex])

[tex]\begin{gathered}

\forall k \in \mathbb{N},\;{\text{what is the smallest }}m \in \mathbb{N}\;{\text{for which}} \hfill \\

\exists \left( {a_i } \right)_{i = 1}^n \;{\text{where }}\forall i > 0,\;a_i \in \left\{ {0,1, \ldots ,m} \right\}, \hfill \\

{\text{such that }}k = \sum\limits_{i = 1}^n {\frac{{a_i }}

{i}} \;\,? \hfill \\

\end{gathered} [/tex]

[tex]\begin{gathered}

\forall k \in \mathbb{N},\;{\text{what is the smallest }}m \in \mathbb{N}\;{\text{for which}} \hfill \\

\exists \left( {a_i } \right)_{i = 1}^n \;{\text{where }}\forall i > 0,\;a_i \in \left\{ {0,1, \ldots ,m} \right\}, \hfill \\

{\text{such that }}k = \sum\limits_{i = 1}^n {\frac{{a_i }}

{i}} \;\,? \hfill \\

\end{gathered} [/tex]

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