Smallest Even Integer whose Euler Phi Value is not the Euler Phi Value of an Odd Integer

Theorem
The smallest even integer whose Euler $\phi$ value is shared by no odd integer is $33 \, 817 \, 088$.

Proof
We have:

Consider the equation:
 * $(1): \quad \map \phi x = 2^{16} \times 257$

Even solutions to $(1)$ are of the form:
 * $x = 2^{9 - \epsilon_0 - 2 \epsilon_1 - 4 \epsilon_2} \times 3^{\epsilon_0} \times 5^{\epsilon_1} \times 17^{\epsilon_2} \times 257^2$

where each $\epsilon_k$ equals $0$ or $1$.

there exists an odd integer $k$ satisfying $(1)$:
 * $\map \phi k = 2^{16} \times 257$

Let $p$ be a prime factor of $k$.

Then as Euler Phi Function is Multiplicative, either:
 * $p - 1 = 2^j$ for some $j \in \Z$ such that $1 \le j \le 16$

or:
 * $p - 1 = 2^j \times 257$ for some $j \in \Z$ such that $1 \le j \le 16$

But $2^j \times 257 + 1$ is composite for $1 \le j \le 16$:

So the only possible prime factors of $k$ are the Fermat primes:
 * $3, 5, 17, 257, 65 \, 537$

such that:
 * $257$ occurs with multiplicity $2$
 * all other prime factors occurs with multiplicity $1$.

Because of the size of $257^2 \times 67 \, 537$ it follows that $257$ and $65 \, 537$ cannot appear together.

We have that:

Thus there can be no odd integer $k$ satisfying $(1)$.

It can be established by computer that $33 \, 817 \, 088$ is the smallest such even integer with this property.