Let \(N_{ARF}(m,\mathscr {C})\) denote the number of Arf numerical semigroups with multiplicity m and conductor \(\mathscr {C}\) . In an earlier paper of the first author (Semigroup Forum 105:478–487, 2022), it was proved that \(N_{ARF}(m,\mathscr {C})\) depends only on the congruence class of \(\mathscr {C}\) modulo m if m is prime. In the same paper, it was noticed that there are composite numbers m for which \(N_{ARF}(m,\mathscr {C})\) depends only on the congruence class of \(\mathscr {C}\) modulo m for some congruence classes, and the author had posed the question of characterizing such m and congruence classes of m for which \(N_{ARF}(m,\mathscr {C})\) is an invariant of those classes. We prove that \(N_{ARF}(m,\mathscr {C})\) is an invariant of the congruence class of \(\mathscr {C}\) (mod m) if and only if \(m = p^n\) and \(\mathscr {C} \equiv (tp^{n-1}+1) \ (\textrm{mod}\ p^n)\) where \(n\in {\mathbb {N}}\) , p is prime, \(t \in \{1, \ldots , p-1\}\) .
