Orders for which there exist exactly six or seven groups

TL;DR

This paper investigates the specific integers n for which the group-counting function g(n) equals exactly 6 or 7, extending previous work that determined such n for values up to 5.

Contribution

It solves the inverse problem of identifying all n with g(n) = 6 or 7, building on prior classifications for smaller values of g(n).

Findings

Identifies all n with g(n) = 6.

Identifies all n with g(n) = 7.

Extends the classification of n for which g(n) takes small values.

Abstract

Much progress has been made on the problem of calculating $g(n)$ for various classes of integers $n$, where $g$ is the group-counting function. We approach the inverse problem of solving the equations $g(n) = 6$ and $g(n) = 7$ in $n$. The determination of $n$ for which $g(n) = k$ has been carried out by G. A. Miller for $1 \le k \le 5$.