Existence of certain integers So the question statement is simple.
Is there any 'large' prime $p$ such that $\phi(p^2)/2$ is odd?
Or weaker, is there any large integer $n$ where $\textbf Z_n^\times$ is cyclic and $\phi(n)/2$ is odd?
I can find out some small cases such as $\phi(7^2) = 2 \cdot 21$, but I cannot find ways to construct big ones nor disprove its existence.
By 'large' I mean, like if you have infinite of them then it can be arbitrarily large.

Expanding on Tobias Kildetoft's comment: given that $\phi(p) = p - 1$ (for $p$ prime) and that $\phi(p^2) = p^2 - p$ it follows that the parity of half $\phi(p^2)$ depends on $p \pmod 4$.
If $p \equiv 1 \pmod 4$, then $p^2 - p \equiv 1 - 1 = 0 \pmod 4$, which means that half $\phi(p^2)$ is still even. But if $p \equiv 3 \pmod 4$, then $p^2 - p \equiv 1 - 3 = 2 \pmod 4$ and half of that is $1 \pmod 4$.
In Wolfram Mathematica, or Wolfram Alpha, try this: Select[Prime[Range[25]], OddQ[EulerPhi[#^2]/2] &] then input the results into Neal Sloane's OEIS. Oops, I forgot about 2. Indeed $\phi(4) = 2$ and half of that is odd. See https://oeis.org/A045326
So the answer is 2 and all primes congruent to 3 modulo 4. Which means that whatever the largest known prime is right now, most likely a Mersenne prime,  is probably a relatively big example of what you're loooking for.
