Abstract :
Yuster (Arithmetic progressions with constant weight, Discrete Math. 224 (2000) 225–237) defines divisor matrices and uses them to derive results on “magic” sequences, i.e. finite sequences a1,a2,…,an with the property that for a certain k all sums ∑j=1kaij with i1,i2,…,ik an arithmetic subsequence of 1,2,…,n, are equal. An important condition is the (conjectured) non-singularity of the elementary divisor matrices Ak, that could only be proved for k with at most two prime divisors. We present a proof for general k, thereby generalizing the results in Yuster [1] (Arithmetic progressions with constant weight, Discrete Math., to appear.). Our exploration of Ak also leads to new proofs, and enables us to add other results, in particular we give the dimension of the space of k-magic sequences of length n for every k and n and over every field.
