A set of integers from 1 through
powers equal the degree-magic constant for each
positive integers .
The case magic series , with magic
constant
(Kraitchik 1942, p. 143). In general, the degree-magic constant is
where harmonic number of order
For example
The numbers of multimagic series of various lengths
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
See also Magic Series
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References Boyer, C. "Multimagic Series." http://multimagie.com/English/Series.htm . Kraitchik, M. "Multimagic Squares." §7.10 in Mathematical
Recreations. Sloane,
N. J. A. Sequences A052456 , A052457 ,
A052458 , and A090037
in "The On-Line Encyclopedia of Integer Sequences." Referenced
on Wolfram|Alpha Multimagic Series
Cite this as:
Weisstein, Eric W. "Multimagic Series."
From MathWorld https://mathworld.wolfram.com/MultimagicSeries.html
Subject classifications
th
order multimagic series is a set of 
distinct integers from 1 through
whose sums of 
th powers equal the degree-
magic constant for each
, 2, ..., 
, where 
and 
are positive integers.
The case 
is a magic series, with magic
constant
magic constant is
is a generalized harmonic number of order 
.
is bimagic since 
and 
.
It is also trimagic since 
. Similarly, 
is trimagic.
are given 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.
