Inverse Totient Function - Maple Help

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NumberTheory

 InverseTotient
 inverse of Euler's totient function

 Calling Sequence InverseTotient(m)

Parameters

 m - positive integer

Description

 • The InverseTotient function returns the inverse image of Euler's totient function.
 • Given a positive integer m, InverseTotient(m) returns the set of integers n that satisfy the equation Totient(n) = m.

Examples

 > $\mathrm{with}\left(\mathrm{NumberTheory}\right):$
 > $\mathrm{InverseTotient}\left(1\right)$
 $\left\{{1}{,}{2}\right\}$ (1)
 > $\mathrm{InverseTotient}\left(8\right)$
 $\left\{{15}{,}{16}{,}{20}{,}{24}{,}{30}\right\}$ (2)
 > $\mathrm{map}\left(\mathrm{Totient},\left\{15,16,20,24,30\right\}\right)$
 $\left\{{8}\right\}$ (3)

There are no numbers besides 1 and 2 with an odd totient.

 > $\mathrm{InverseTotient}\left(3\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{union}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{InverseTotient}\left(2347\right)$
 ${\varnothing }$ (4)
 > $\mathrm{union}\left(\mathrm{seq}\left(\mathrm{InverseTotient}\left(i\right),i=3..99,2\right)\right)$
 ${\varnothing }$ (5)

The following command plots the points (m, n), where n is a member of InverseTotient(m), for m from $1$ to $1000$.

 > $\mathrm{plots}:-\mathrm{pointplot}\left(\mathrm{union}\left(\mathrm{seq}\left(\mathrm{map}\left(n↦\left[m,n\right],\mathrm{InverseTotient}\left(m\right)\right),m=1..1000\right)\right),\mathrm{labels}=\left["m","InverseTotient\left(m\right)"\right],\mathrm{labeldirections}=\left[\mathrm{horizontal},\mathrm{vertical}\right],\mathrm{color}="Fuchsia",\mathrm{symbol}=\mathrm{circle}\right)$

Compatibility

 • The NumberTheory[InverseTotient] command was introduced in Maple 2016.