Some Observations about the "Generalized Abundancy Index"

Abstract

Let $\mathcal{A}(\ell ,n)\subset S_n^{\ell }$ denote the set of all $\ell$-tuples $(\ddot{I}{€}_1,\dots ,\ddot{I}{€}_{\ell })$, for $\ddot{I}{€}_1,\dots ,\ddot{I}{€}_{\ell }\in S_n$ satisfying: $\forall i<j$ we have $\ddot{I}{€}_i\ddot{I}{€}_j=\ddot{I}{€}_j\ddot{I}{€}_i$. Considering the action of $S_n$ on $[n]=\{1,\dots ,n\}$, let $\hat{I}º(\ddot{I}{€}_1,\dots ,\ddot{I}{€}_{\ell })$ be equal to the number of orbits of the action of the subgroup $⟨\ddot{I}{€}_1,\dots ,\ddot{I}{€}_{\ell }⟩\subset S_n$. There has been interest in the study of the combinatorial numbers $A(\ell ,n,k)$ equal to the cardinalities $|\{(\ddot{I}{€}_1,\dots ,\ddot{I}{€}_{\ell })\in \mathcal{A}(\ell ,n):\hat{I}º(\ddot{I}{€}_1,\dots \ddot{I}{€}_{\ell })=k\}|$. If one defines $B(\ell ,n)=A(\ell ,n,1)/(n-1)!$, then it is known that $B(\ell ,n)={\sum }_{(f_1,\dots ,f_{\ell })\in {\mathbb{N}}^{\ell }}{1}_{\{n\}}(f_1\cdots f_{\ell }){\prod }_{r=1}^{\ell -1}{f}_r^{\ell -r}$. A special case, $\ell =2$, is $B(2,n)={\sum }_{d|n}d=\ddot{I}{ƒ}_1(n)$ the sum-of-divisors function. Then $A(2,n,1)/n!=B(2,n)/n$ is called the abundancy index: $\ddot{I}{ƒ}_1(n)/n$. We call $B(\ell ,n)n^{-\ell +1}$ the ``generalized abundancy index.'' Building on work of Abdesselam, using the probability model, we prove that ${\lim }_{N\to \infty }{N}^{-1}{\sum }_{n=1}^{N}B(\ell ,n)n^{-\ell +1}$ equals $\hat{I}\P (2)\cdots \hat{I}\P (\ell )$. Motivated by this we state a more precise conjecture for the asymptotics of $-\hat{I}\P (2)+N^{-1}{\sum }_{n=1}^N(B(2,n)/n)$.

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