= Solution
The <Euler totient function> $\phi(n)$ is the number of residue classes modulo $n$ that are coprime to $n$. The <Chinese remainder theorem> says that for pairwise coprime moduli $m_i$, reduction induces a bijection
$$
\mathbb Z/(m_1\cdots m_r)\mathbb Z
\longrightarrow\prod_{i=1}^r\mathbb Z/m_i\mathbb Z.
$$
It restricts to a bijection of unit groups. If $n=p_1\cdots p_r$ is squarefree, then
$$
\boxed{\phi(n)=\prod_{i=1}^r(p_i-1)}.
$$
Each of the primes $3,5,17,257$ is one more than a power of two, so every squarefree product of them has power-of-two totient. Ten odd examples are
$$
\boxed{3,5,15,17,51,85,255,257,771,1285}.
$$
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