Cyclic group is Leinster group if and only if of perfect order

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Statement

The cyclic group of order $n$ is a Leinster group if and only if $n$ is a perfect number, that is, a number such that $2n=\sum _{d\mid n}d$.

Proof

The subgroups of the cyclic group of order $n$ consist of a unique cyclic subgroup of order $d$, for any $d\mid n$.

Thus $\sum _{H\leq C_{n}}|H|=\sum _{d\mid n}d=2n$ if and only if $n$ is perfect. Hence, the cyclic group of order $n$ is a Leinster group if and only if $n$ is perfect.

List of small cyclic groups which are Leinster groups

The perfect numbers are OEIS:A000396: 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, ...

So the cyclic Leinster groups are cyclic group:Z6, cyclic group:Z28, cyclic group:Z496, cyclic group:Z8128, ...

Remarks

Note a Leinster group needs not be cyclic, for example Dicyclic group:Dic12 is a Leinster group.