**I.** Prime $n$ seems a necessary condition for prime $g(n)$, not only for $f(n)=3^n+5^n+7^n$, $g(n)=\frac{3^n+5^n+7^n}{15}$, but also for similar cases.

E.g. taking the trio of odd integers less than the posted case, for $f(n)=1^n+3^n+5^n$, with$$g(n)=\frac{1^n+3^n+5^n}{3}$$ and $n<100$, $g(n)$ is prime only for prime $n=5, 13, 17, 29, 41$: $$1123$$$$407432483$$$$ 254356197763$$$$62088194517824356003$$$$15158245041706469424307579843$$The next prime $g(n)$ is for prime $n=113$.

Likewise for some increasing trios of consecutive odd integers.$$\begin{array}{cccc}\text{f(n)} & \text{g(n)} & n & \text{prime g(n)}\\ \hline5^n+7^n+9^n & \frac{f(n)}{21} & 5 & 3761\\- & - & 43 & 5131181926598620870059044278576189362457\\
7^n+9^n+11^n & \frac{f(n)}{9} & 1 & 3\\- & - & 11 & 35407784107\\- & - & 13 & 4129051886963\\- & - & 17 & 58039648968105283\\9^n+11^n+13^n & \frac{f(n)}{33} & 5 & 17921\\- & - & 7 & 2636929\\- & - & 29 & 6155428662438784503568282263041\\- & - & 41 & 142437341620935541845482699492428865265971201\\11^n+13^n+15^n & \frac{f(n)}{39} & 11 & 275057126257\\- & - & 37 & 844287359404869502215865700763611372145761\\- & - & 67 & ---\\\end{array}$$For the last $n=67$, prime $g(n)$=$$ 161094059489831593214989598447345339537628 222953633116135188147969918645655217$$

**II.** But taking **non**-consecutive odd integers, e.g. $f(n)=1^n+3^n+7^n$, where $g(n)=f(n)$ since $1+3+7=11$ is not divisible by $3$, we find $g(n)$ is prime for $n=1, 13, 17$, but also for composite $n=49$.

Again, taking three consecutive **integers** instead of three consecutive **odd** integers, composite $n$ can yield prime $g(n)$. E.g. for $f(n)=1^n+2^n+3^n$, where $g(n)=\frac{f(n)}{12}$ for odd $n>3$, we get prime $g(n)$:$$3$$$$23$$$$193$$$$133543$$$$10772603$$$$7845963953$$$$ 858420955076202578510080530923$$$$176741262253776179925474237053751128366837273$$when $n=3, 5, 7, 13, 17, 23, 65, 95$.

**III.** If prime $g(n)$ for composite $n$ are as elusive for the trios of consecutive odd integers, sampled above for odd $n<100$, as they are found to be for the posted $\frac{3^n+5^n+7^n}{15}$, can the conjecture then be more generally put: If $f(n)$ is the sum of like odd powers of three consecutive odd integers, and $g(n)$ is $f(n)$ cleared of factors common to all $f(n)$, and $g(n)$ is prime, then $n$ is prime?

However, this is not universally true. In the posted case the three consecutive odd integers are all prime, and in the nearby cases considered above at least one of the integers is prime. Probing the possible relevance of this by examining $g(n)$ for the smallest trio of consecutive odd integers all composite:$$g(n)=\frac{91^n+93^n+95^n}{93}$$we find prime $g(35)$=$$30301372938052318586648696678367793606476198590474743450180085922931$$So if $g(n)$ is indeed never prime for $\frac{3^n+5^n+7^n}{15}$, then the cause seems to lie in something more specific than the behavior of like powers of consecutive odd integers.