A set of
distinct numbers taken from the interval form a magic series if their sum is the th magic constant

(Kraitchik 1942, p. 143). The numbers of magic series of orders , 2, ..., are 1, 2, 8, 86, 1394, ... (OEIS A052456).
The following table gives the first few magic series of small order.

magic series

1

2

,

3

, , , , , , ,

If the sum of the th
powers of these number is the magic constant of
degree
for all ,
then they are said to form a th order multimagic series.
Here, the magic constant
of degree
is defined as
times the sum of the first th
powers,

Kraitchik, M. "Magic Series." §7.13.3 in Mathematical
Recreations. New York: W. W. Norton, pp. 143 and 183-186,
1942.Sloane, N. J. A. Sequence A052456
in "The On-Line Encyclopedia of Integer Sequences."