TITLE: Density of multi-grade solutions to $x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k$ for $k = 5$ or $6$?
QUESTION [5 upvotes]: Given the Diophantine equation,
$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k\tag1$$
there is the rather curious observation that the smallest positive solutions for $k=5$ or $6$ is multi-grade.
$$24^k+28^k+67^k=3^k+54^k+62^k,\quad k = 1,5$$
$$15^k + 10^k + 23^k = 3^k + 19^k + 22^k,\quad k = 2,6$$
Duncan Moore has exhaustively searched $(1)$ for all positive and primitive solutions below a bound $Z$. Table 1 is for $k=5$, while Table 2 is for $k=6$. We summarize the data below.
I. Table 1:
$$\begin{array}{|c|c|c||}
\text{# of solns}&\color{blue}{A:=\text{(% of}\; k = 1,5)}&\text{diff}\\
2^0\cdot168&63.7\text{%}& \\
2^1\cdot168&65.8\text{%}&+2.7\\
2^2\cdot168&65.6\text{%}&-0.3 \\
2^3\cdot168&63.6\text{%}&-2.0\\
2^4\cdot168&61.0\text{%}&-2.6\\
2^5\cdot168&59.1\text{%}&-1.9\\
\end{array}$$
Note: To address one comment below, $A$ is the percentage of solns given in the first column that is valid for both $k=1,5$. For example, out of the first $2^5\cdot168 = 5376$ solns, then $59.1\text{%}$ are for $k=1,5$.
Each row doubles the $\text{#}$. Since Moore's database has $5393$ solns, and $5393/2^5\approx168.53$, then I used that as the base value. 
II. Table 2:
$$\begin{array}{|c|c|c|}
\text{# of solns}&\color{blue}{B:=\text{(% of}\; k = 2,6)}&\text{diff}\\
50&80\text{%}& \\
100&85\text{%}&+5.0\\
200&89\text{%}&+4.0\\
400&91.7\text{%}&+2.7\\
\end{array}$$
Note: Thus, out of the first $400$ solns, then a whopping $91.7\text{%}$ of them are actually multi-grade for $k=2,6$. (I'm not sure if excluding non-primitive solutions below the bound $Z$ is relevant. Program-wise, it seems easier to just include them.)
Questions:

Why is the percentage of $A$ decreasing, while that of $B$ is apparently increasing? Or will $B$ eventually have a negative diff like $A$? (The data is too small to be conclusive.)
If both are decreasing, will $A,B \to 0$? Or will it taper off to some constant? 

P.S. This answer to a related post might be informative. Incidentally, the smallest solutions to,
$$x_1^k+x_2^k+x_3^k+x_4^k = y_1^k+y_2^k+y_3^k+x_4^k\tag2$$
are also multigrades as $k=1,5$, and $k=2,6$, though there are no exhaustive tables for these.

REPLY [2 votes]: Not an answer, just a graphic illustrating the percentages $A$ and $B$ of multi-grade solutions vs. all solutions, sorted according to the biggest contained term. It is hard to believe they should drop to $0$ as $n$ grows, especially $B$, but sure enough, the available data after all only cover a tiny part of infinity...