Characterization conjecture for equality in the bound on g(n)

Let $g(n)$ be the quantity associated with corresponding sequences, and suppose the established upper bound is $g(n)\leq 2n$.

Equality characterization conjecture. The upper bound is attained,

$$g(n)=2n,$$

if and only if $n$ is prime or $n=6$.

The conjecture follows results showing that the bound $g(n)\leq 2n$ is sharp, and proposes that equality occurs only for prime inputs and the exceptional value $n=6$.