ON THE REPRESENTATION OF INTERGERS AS SUMS OF DISTINCT TERMS FROM A FIXED SEQUENCE,

Consideration is given a problem that has received considerable attention recently. Let a sub 1, a sub 2, a sub 3, . . . be a sequence of positive integers. If every sufficiently large integer can be represented as a sum of distinct terms from this sequence, one says that the sequence is complete. The general problem is Characterize complete sequences. This memorandum considers this problem for the class of sequences which are either increasing with a sub n 0n alpha or strictly increasing with a sub n 0n 1alpha, where 0 alpha 1. It shows that a necessary and sufficient condition for such a sequence to be complete is that at least one term from every infinite arithmetic progression should be representable as a sum of distinct terms from the sequence. Author