Counting monogenic monoids and inverse monoids

In this short note, we show that the number of monogenic submonoids of the full transformation monoid of degree n for n > 0, equals the sum of the number of cyclic subgroups of the symmetric groups on 1 to n points. We also prove an analogous statement for monogenic subsemigroups of the finite full transformation monoids, as well as monogenic inverse submonoids and subsemigroups of the finite symmetric inverse monoids.