What is the maximum value of $(a+ b+c)$ if $(a^n + b^n + c^n)$ is divisible by $(a+ b+c)$ where the remainder is 0?
The answer is indeed $6$. Here is a complete solution.
First, take a prime $p$ such that, $p\mid a+b+c$. Note that, if $p$ divides two of $a$ and $b$, then it must divide the third, contradicting with $(a,b,c)=1$. Thus, $p$ either divides none of them, or only one.
- If $p\nmid a,b,c$, then by taking $n=p-1$, we have, by Fermat's theorem that $a^{p-1}\equiv b^{p-1}\equiv c^{p-1}\equiv 1 \pmod{p}$. Thus, $p\mid 3$, hence $p=3$.
- If $p\mid a$, and $p\nmid b,c$, then we obtain $p\mid 2$, by taking $n=p-1$.
Therefore, only prime divisors of $a+b+c$ are $2$ or $3$. Now, for $n=2$, we get using $(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)$ that $a+b+c\mid 2(ab+bc+ca)$. Also, for $n=3$, using $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$, we have $a+b+c\mid 3abc$. Next, note that, if $9\mid a+b+c$, then $3\mid abc$. Hence, either $a$ or $b$ or $c$ is divisible by $3$. Now, if $3\mid a$, then using $a+b+c\mid 2(ab+bc+ca)$, we obtain that $3\mid bc$, hence $3\mid b$ or $3\mid c$. However, this, together with $3\mid a+b+c$ contradicts with $(a,b,c)=1$. Hence, $9\nmid a+b+c$.
Similarly, if $4\mid a+b+c$, then note that among $a,b,c$ exactly one is even, suppose it is $a$, that is, $4\mid a$. But this gives, $4\mid 2(ab+bc+ca)$, yielding $2\mid (ab+bc+ca)$, yielding $2\mid bc$. However, since $b$ and $c$ are odd, this is a clear contradiction.
Thus, $a+b+c=3$ or $a+b+c=6$. In former, we get $(1,1,1)$, which is really $n$-good for any $n$. For the latter, we have $(3,2,1)$ or $(4,1,1)$. For the former, $(3,2,1)$ is not $n-$ good for any even $n$, using modulo $3$. For the latter, it is easy to see that it is $n$-good for any $n$.
Hence, we are done.
Note (Alternative) There is an alternative way of proving that $a+b+c$ can only admit $2$ or $3$ as its prime divisors. To see this, suppose $p>3$ divides $a+b+c$. Then, $p\mid 3abc$ implies $p\mid abc$. Using $(a,b,c)=1$, we see that exactly one of $a,b,c$ is divisible by $p$. Suppose, it is $a$. Then, $p\mid ab+bc+ca$, together with $p\mid ab+ac$ implies $p\mid bc$, which clearly is a contradiction. From here, one can finish in exact same way as in above proof, i.e., prove $4,9\nmid a+b+c$, and finish.
The question is to find the largest sum of $a, b, c$, given they're all relatively prime to each other and divide
$$a^n + b^n + c^n \text{ } \forall \text{ } n \ge 1 \tag{1}\label{eq1}$$
Since $a, b, c \ge 1$, then $a + b + c \ge 3$, so it consists of one or more prime factors. Call one of these prime factors $d$. Thus,
$$a + b + c \equiv a^n + b^n + c^n \equiv 0 \pmod d \tag{2}\label{eq2}$$
As this must hold for all $n \ge 1$, consider $n = 3$ and substitute
$$c \equiv -a - b \pmod d \tag{3}\label{eq3}$$
into the middle part of the congruence in \eqref{eq2} to get
$$a^3 + b^3 + \left(-a - b\right)^3 \equiv 0 \pmod d \tag{4}\label{eq4}$$
This simplifies to
$$-3ab^2 - 3a^2b = -3ab\left(a + b\right) \equiv 0 \pmod d \tag{5}\label{eq5}$$
Thus, $d$ must divide $3$, $a$, $b$ and/or $a + b \equiv -c \pmod d$. If $d$ is not $3$, then note that if it must divide just one of $a$, $b$ and $-c$. This is because if it divides any $2$, say $a$ and $b$, it must also divide $c$. As the equations are symmetric in $a, b \text{ and } c$, WLOG, assume that $d$ divides $a$, i.e., $a \equiv 0 \pmod d$. Thus, $b + c \equiv 0 \pmod d$, i.e., $b \equiv -c \pmod d$. Now, use $n = 2$ in the congruence in the middle part of \eqref{eq2} to get
$$a^2 + b^2 + c^2 \equiv 0 + \left(-c\right) + c^2 \equiv 2c^2 \equiv 0 \pmod d \tag{6}\label{eq6}$$
As $d$ doesn't divide $c$, this means that $d$ must divide $2$, i.e., it must be $2$. Thus, the only possible prime factors of $a + b + c$ are $3$, as mentioned earlier, and $2$ as shown here.
Based on what I have found here, I believe the maximum value is just the product of those $2$ factors, i.e., $6$, which can be obtained from
$$\left(a, b, c\right) = \left(1, 1, 4\right) \tag{7}\label{eq7}$$
You can easily verify that $a^n + b^n + c^n = 1 + 1 + 4^n \equiv 0 \pmod 6$ for all $n \ge 1$ as $1 + 1 = 2$, so $2$ divides it, and $4^n \equiv 1^n \equiv 1 \pmod 3$, so $3$ divides it as well. I am posting this partial solution for now so you have something to start from, as I believe I'm on the right track. I need to go for a while, but I will keep thinking about this and, if I determine the rest of the solution, I will add it later.