Let $gnu(n)$ be the number of groups of order $n$. If $n$ is cubefree, (there is no prime $p$ with $p^3|n$), does the inequality $gnu(n)<n$ always hold for $n>1$ ?
According to GAP, upto $50,000$, this is the case. Since the maximal gnu I got was $3093$, I conjecture that the inequality holds for all cubefree $n$. Can this be proven ?
For squarfree $n>1$, it is known that $gnu(n)\le \phi(n)<n$.