2-adic stratification of totients (Q2115173)

Let \(\mathcal{V}\) denote the set of the values taken by Euler's totient function \(\varphi\): \[ \varphi(n)=\#\{1\le a\le t n: \gcd(a,n)=1\}, \] and let \[ \mathcal{V}^l= \{m\in \mathcal{V}:m\equiv 2^l\bmod 2^{l+1}\}. \] The authors of the paper examine problems related to the structure of sets \(\mathcal{V}\) and \(\mathcal{V}^l, l\ge 1\). Particular attention is paid to the number of elements in the subsets of these sets.