A magic series of order 
is a set of 
distinct positive integers
not exceeding 
whose 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 numbers is the magic constant of
degree 
for all 
,
2, ..., 
,
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,
 |
where 
is a harmonic number of order 
.
