Show that there is no integer n with $\phi(n)$ = 14

If a prime $p\mid n$, then $p-1\mid\phi(n)$. If $\phi(n)=14$, then, since the divisors of $14$ are $\{1,2,7,14\}$, $p\in\{2,3\}$. Thus, $n=2^a3^b$ and $\phi(n)$ is of the form $2^j3^k$. However, there is no factor of $7$ in $2^j3^k$.

We answer the question, but in a way that is far too convoluted for the particular numerical example of the post. We prove the following result.

Lemma: If $\varphi(n)=2q$, where $q$ is an odd prime, then $2q+1$ is prime.

An immediate consequence is that we cannot have $\varphi(n)=14$. For then $q=7$, but $2q+1$ is not prime.

We now prove the lemma. Suppose that $\varphi(n)=2q$. We divide the analysis into cases.

Case $1$: Maybe $n=2^km$, where $k\ge 3$ and $m$ is odd. Then by the multiplicativity oof $\varphi$, we have $\varphi(n)=\varphi(2^k)\varphi(m)$. This is impossible, since $\varphi(2^k)=2^{k-1}$, which is divisible by $4$. But $2q$ is not divisible by $4$.

Case $2$: Maybe $n=4m$, where $m$ is odd. Then $\varphi(n)=2\varphi(m)$. Since $2q$ is twice an odd number, it follows that $\varphi(m)$ is odd. This is only possible for an odd number $m$ if $m=1$. But then $\varphi(m)=2\ne 2q$.

Case $3$: Maybe $n=2m$ or $n=m$, where $m$ is an odd number. In either case, $\varphi(n)=\varphi(m)$.

Suppose that $m$ can be factored as a product $st$, where $s$ and $t$ are relatively prime, and neither $s$ nor $t$ is equal to $1$. Then $\varphi(s)$ and $\varphi(t)$ are each even, so $\varphi(m)$ is divisible by $4$. But $2q$ isn't, so we cannot have $\varphi(m)=2q$.

It remains to deal with the case where $m$ is a prime power, perhaps equal to $1$. If $m=1$, then $2q=2$, not twice an odd prime.

Now suppose that $m=p^k$, where $p$ is an odd prime, and $k\ge 1$. Then $\varphi(m)=p^{k-1}(p-1)$. This can be twice an odd prime in $2$ ways: (i) $k=1$ and $p-1$ is twice an odd prime, or (ii) $p=3$ and $k=2$.

In Case (i), $2q=p-1$, so $2q+1$ is prime. In Case (ii), $2q=6$, so again $2q+1$ is prime.