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Omni-MATH / Find all functions f:N→N such that the following conditions are true for every pair of positive integers (x,y): (i): x and f(x) have the same number of positive divisors. (ii): If …

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problem

Find all functions f:N→Nf : \mathbb{N} \rightarrow \mathbb{N} such that the following conditions are true for every pair of positive integers (x,y)(x, y): (i)(i): xx and f(x)f(x) have the same number of positive divisors. (ii)(ii): If x∤yx \nmid y and y∤xy \nmid x, then: gcd(f(x),f(y))>f(gcd(x,y))\gcd(f(x), f(y)) > f(\gcd(x, y))
Plain-text mathematical notation (without MathML)
Find all functions f:N→N such that the following conditions are true for every pair of positive integers (x,y):
(i): x and f(x) have the same number of positive divisors.
(ii): If x∤y and y∤x, then:
gcd(f(x),f(y))>f(gcd(x,y))
Original LaTeX notation
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that the following conditions are true for every pair of positive integers $(x, y)$:
$(i)$: $x$ and $f(x)$ have the same number of positive divisors.
$(ii)$: If $x \nmid y$ and $y \nmid x$, then:
$$\gcd(f(x), f(y)) > f(\gcd(x, y))$$

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Source and history

task_id
640b8dda-d22c-5bfb-aa8d-9f9fcb9a0745
task_key
test--640b8dda-d22c-5bfb-aa8d-9f9fcb9a0745
task_revision_id
1
config / split
/ test

Official source

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