Multimagic Series

$n$ numbers form a $p$-multimagic series if the sum of their $k$th powers is the Magic Constant of degree $k$ for every $k=1$, ..., $p$. The following table gives the number of $p$-multimagic series $N_p$ of given orders $n$ (Kraitchik 1942).

$n$	$N_1$	$N_2$	$N_3$
2	2
3	8
4	86	2	2
5	1,394	8	2
6	0	98	0
7	0	1,844	0
8	0	38,039	115
9	0	0	41
10	0	0	0
11	0	0	961

References

Kraitchik, M. ``Multimagic Squares.'' §7.10 in Mathematical Recreations. New York: W. W. Norton, pp. 176-178, 1942.