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Omni-MATH / Denote Z_(>0)={1,2,3,...} the set of all positive integers. Determine all functions f:Z_(>0)→Z_(>0) such that, for each positive integer n, hspace{1cm}i) sum_{k=1}^{n}f(k) is a per…

Problem

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problem

Denote Z>0={1,2,3,...}\mathbb{Z}_{>0}=\{1,2,3,...\} the set of all positive integers. Determine all functions f:Z>0→Z>0f:\mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0} such that, for each positive integer nn, $\hspace{1cm}i) \sum_{k=1}^{n}f(k)$ is a perfect square, and $\vspace{0.1cm}$ $\hspace{1cm}ii) f(n)$ divides n3n^3.
Plain-text mathematical notation (without MathML)
Denote Z_(>0)={1,2,3,...} the set of all positive integers. Determine all functions f:Z_(>0)→Z_(>0) such that, for each positive integer n,
$\hspace{1cm}i) \sum_{k=1}^{n}f(k)$ is a perfect square, and
$\vspace{0.1cm}$
$\hspace{1cm}ii) f(n)$ divides n³.
Original LaTeX notation
Denote $\mathbb{Z}_{>0}=\{1,2,3,...\}$ the set of all positive integers. Determine all functions $f:\mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0}$ such that, for each positive integer $n$,
$\hspace{1cm}i) \sum_{k=1}^{n}f(k)$ is a perfect square, and
$\vspace{0.1cm}$
$\hspace{1cm}ii) f(n)$ divides $n^3$.

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Source and history

task_id
ca715248-0418-5624-9a88-eb64a6d433ca
task_key
test--ca715248-0418-5624-9a88-eb64a6d433ca
task_revision_id
1
config / split
/ test

Official source

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