Number of distinct prime factors of $a^n+b^n+c^n$ This question is a generalization of this other one. 
Problem: Given a constant $k$ and distinct positive integers $a,b,c$, prove that there exist an integer $n>1$ for which the number of distinct prime factors of $a^n+b^n+c^n$ is at least $k$.
Moreover, do you know if there exists some quantitative bound of this type with order, let's say, greater than $\ln\ln\ln n$? I mean, if $a^n+b^n+c^n$ would behave really randomly, then by the Erdos-Kac theorem $\omega(a^n+b^n+c^n)$ would have normal order $\ln \ln (a^n+b^n+c^n) \gg \ln n$..

Let denote te phrase as $A=a^n+b^n+c^n$. It can be shown that if $a=x-y$, $b=y-z$ and $c=z-x$ then A can be factorized as follows:
If $n=6k-1$, then:
$(x-y)^n+(y-z)^n+(z-x)^n=n(x-y)(y-z)(z-x)(x^2+y^2+z^2-xy-yz-xz)$
If $n=6k+1$, then:
$(x-y)^n+(y-z)^n+(z-x)^n=n(x-y)(y-z)(z-x)(x^2+y^2+z^2-xy-yz-xz)^2$
Now following cases can be considered:
1- n is prime and a, b and c are co-primes; considering the fact that $x-y$ and other factors are even , then  A has at least two prime factors; i.e. $n$ and $2$.
2- If n is not prime ; $n=d^{\alpha}.e^{\beta}. . . $, then A has at least $t+1$ distinct primes, where t is number of primes in n.
3- if factors are also composite, in addition to being even, then number of distinct primes is greater than $t+1$, where t is number of primes in n.
