bimagic series | trimagic series

A set n distinct numbers taken from the interval [1, n^2] form a magic series if their sum is the nth magic constant M_n = 1/2 n(n^2 + 1) (Kraitchik 1942, p. 143). If the sum of the kth powers of these numbers is the magic constant of degree k for all k element [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/n sum_(i = 1)^(n^2) i^k = H_(n^2)^(-k)/n, where H_n^(k) is a generalized harmonic number of order k.