The enumeration of plane (3,6)-triangulations and the form of odd perfect numbers

Abstract

Let $\mathcal{P}$ be the family of all $3$-connected plane triangulations with vertices of degree $3$ or $6$. The number $d(n)$ of non-isomorphic triangulations belonging to $\mathcal{P}$ of degree $2n+2$ are enumerated, for $n⩾1$. From the formula for $d(n)$ (where $n$ is odd and $n>1$), we obtain a strengthening of Euler's theorem regarding the form of odd perfect number.

14 pages, 4 figures