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Magic Series

A set of n distinct numbers taken from the interval [1,n^2] form a magic series if their sum is the nth magic constant

M_n=1/2n(n^2+1)

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

nmagic series
1{1}
2{1,4}, {2,3}
3{1,5,9}, {1,6,8}, {2,4,9}, {2,5,8}, {2,6,7}, {3,4,8}, {3,5,7}, {4,5,6}

If the sum of the kth powers of these number is the magic constant of degree k for all k in [1,p], then they are said to form a pth order multimagic series. Here, the magic constant M_n^((j)) of degree k is defined as 1/n times the sum of the first n^2 kth powers,

M_n^((k))=1/nsum_(i=1)^(n^2)i^k=(H_(n^2)^((-p)))/n,

where H_n^((k)) is a harmonic number of order k.

See also

Magic Constant, Magic Square, Multimagic Series

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References

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."

Referenced on Wolfram|Alpha

Magic Series

Cite this as:

Weisstein, Eric W. "Magic Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MagicSeries.html

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