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.

$\mathrm{with}\left(\mathrm{NumberTheory}\right)\:$

$\mathrm{InverseTotient}\left(1\right)$

$\left\{1\,2\right\}$

$\mathrm{InverseTotient}\left(8\right)$

$\left\{15\,16\,20\,24\,30\right\}$

$\mathrm{map}\left(\mathrm{Totient}\,\left\{15\,16\,20\,24\,30\right\}\right)$

$\left\{8\right\}$

$\mathrm{InverseTotient}\left(3\right)\mathbf{union}\mathrm{InverseTotient}\left(2347\right)$

$\varnothing$

$\mathrm{`union`}\left(\mathrm{seq}\left(\mathrm{InverseTotient}\left(i\right)\,i=3..99\,2\right)\right)$

$\varnothing$

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

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

For more information on Maple 2016 changes, see Updates in Maple 2016.