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Omni-MATH / For every positive integer n with prime factorization n=∏_(i=1)^(k)p_(i)^(α_(i)), define mho(n) = sum_{i: ; p_i > 10^{100}} alpha_i. That is, mho(n) is the number of prime factors …

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problem

For every positive integer nn with prime factorization n=∏i=1kpiαin = \prod_{i = 1}^{k} p_i^{\alpha_i}, define \[\mho(n) = \sum_{i: \; p_i > 10^{100}} \alpha_i.\] That is, $\mho(n)$ is the number of prime factors of nn greater than 1010010^{100}, counted with multiplicity. Find all strictly increasing functions f:Z→Zf: \mathbb{Z} \to \mathbb{Z} such that \[\mho(f(a) - f(b)) \le \mho(a - b) \quad \text{for all integers } a \text{ and } b \text{ with } a > b.\] [i]
Plain-text mathematical notation (without MathML)
For every positive integer n with prime factorization n=∏_(i=1)^(k)p_(i)^(α_(i)), define
\[\mho(n) = \sum_{i: \; p_i > 10^{100}} \alpha_i.\]
That is, $\mho(n)$ is the number of prime factors of n greater than 10¹⁰⁰, counted with multiplicity.

Find all strictly increasing functions f:Z→Z such that
\[\mho(f(a) - f(b)) \le \mho(a - b) \quad \text{for all integers } a \text{ and } b \text{ with } a > b.\]

[i]
Original LaTeX notation
For every positive integer $n$ with prime factorization $n = \prod_{i = 1}^{k} p_i^{\alpha_i}$, define
\[\mho(n) = \sum_{i: \; p_i > 10^{100}} \alpha_i.\]
That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity.

Find all strictly increasing functions $f: \mathbb{Z} \to \mathbb{Z}$ such that
\[\mho(f(a) - f(b)) \le \mho(a - b) \quad \text{for all integers } a \text{ and } b \text{ with } a > b.\]

[i]

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Source and history

task_id
31c72276-398f-5033-b786-d80402eb5bd5
task_key
test--31c72276-398f-5033-b786-d80402eb5bd5
task_revision_id
1
config / split
/ test

Official source

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