Multimagic Series

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

(Kraitchik 1942, p. 143). If the sum of the th powers of these numbers 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,

where is a generalized harmonic number of order .

For example is bimagic since and . It is also trimagic since . Similarly, is trimagic.

The numbers of magic series of various lengths are gives in the following table for small orders (Kraitchik 1942, p. 76; Boyer), where the , 3, and 4 values were corrected and extended by Boyer and Trump in 2002.

Sloane	A052456	A052457	A052458	A090037
1	1	1	1	1
2	2	0	0	0
3	8	0	0	0
4	86	2	2	0
5	1394	8	2	0
6	32134	98	0	0
7	957332	1844	0	0
8	35154340	38039	121	0
9	1537408202	949738	126	0
10	78132541528	24643236	0	0
11	4528684996756	947689757	31187	0
12	295011186006282		2226896	106