For every integer k≥1 whose binary expansion contains a bit i such that the Fermat number F_i=2^(2^i)+1 is composite, the smallest n with φ(n)=2^k is exactly 2^(k+1), i.e. n_a/a=2 for a=2^k. Since F_5=641·6700417 is composite, this holds for the infinite set of k with bit 5 set; hence there are infinitely many totient values a with n_a/a exactly 2, and limsup over totient values of n_a/a is ≥ 2. Complement: if every bit i of k has F_i prime, then n_{2^k}=∏_{i∈B(k)}F_i ∈ (2^k, 2^(k+1)), ratio <2. Unconditionally n_{2^k}=2^(k+1) for every 32≤k<2^33.