Open-problem statements, with their sources attached.

Open benchmark statement: does an $\mathrm{AME}(7,10)$ state exist?

Let $σ_1(n)=σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$. Is it true that, for every $m, n ≥ 2$, there exist some $i, j$ such that $σ_i(m) = σ_j(n)$?

Open benchmark statement: does an $\mathrm{AME}(8,4)$ state exist?

Are there infinitely many barriers for ω?

Open benchmark statement: does an $\mathrm{AME}(8,6)$ state exist?

Erdős believed there should be infinitely many barriers for Ω, the total prime multiplicity.

Open benchmark statement: does an $\mathrm{AME}(8,10)$ state exist?

Does there exist some ε > 0 such that there are infinitely many ε-barriers for ω?

Open benchmark statement: does an $\mathrm{AME}(9,6)$ state exist?

Let $h_1(n) = h(n)$ and $h_k(n) = h(h_{k-1}(n))$. Is it true, for any $m,n$, there exist $i$ and $j$ such that $h_i(m) = h_j(n)$?

Open benchmark statement: does an $\mathrm{AME}(9,10)$ state exist?

Let V(x) count the number of n≤x such that ϕ(m)=n is solvable. Does V(2x)/V(x)→2 ?

Open benchmark statement: does an $\mathrm{AME}(10,6)$ state exist?

Let V(x) count the number of n≤x such that ϕ(m)=n is solvable. Is there an asymptotic formula for V(x)?

Open benchmark statement: does an $\mathrm{AME}(10,10)$ state exist?

Let$V'(x)=\#\{\phi(m) : 1\leq m\leq x\}$and$V(x)=\#\{\phi(m) \leq x : 1\leq m\}.$ Does $\lim V(x)/V'(x)$ exist?

Formalization note: We formalize the limit of the inverse fraction V'(x)/V(x) to ensure the limit is finite (bounded between 0 and 1).

Open benchmark statement: does an $\mathrm{AME}(11,3)$ state exist?

Is it $>1$?

Open benchmark statement: does an $\mathrm{AME}(11,4)$ state exist?

It is open whether the set of non-cototients has positive density.