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Omni-MATH / Find all functions f:Z^(+)→Z^(+) such that for all positive integers m,n with m≥n, f(mφ(n³))=f(m)⋅φ(n³). Here φ(n) denotes the number of positive integers coprime to n and not exce…

Problem

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problem

Find all functions

f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+

such that for all positive integers

m,n

with

m\ge n

f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3).

Here

\varphi(n)

denotes the number of positive integers coprime to

n

and not exceeding

n

Plain-text mathematical notation (without MathML)

Find all functions f:Z^(+)→Z^(+) such that for all positive integers m,n with m≥n, f(mφ(n³))=f(m)⋅φ(n³).
Here φ(n) denotes the number of positive integers coprime to n and not exceeding n.

Original LaTeX notation

Find all functions $f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for all positive integers $m,n$ with $m\ge n$, $$f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3).$$
Here $\varphi(n)$ denotes the number of positive integers coprime to $n$ and not exceeding $n$.

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task_id: 85f54d7d-59ea-574e-ae32-440ea23e812c
task_key: test--85f54d7d-59ea-574e-ae32-440ea23e812c
task_revision_id: 3
config / split: / test

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