On the representation of integers as sums of distinct terms from a fixed sequence

by Jon H. Folkman

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Consideration of a problem that has received considerable attention recently. Given a sequence of positive integers, if every sufficiently large integer can be represented as a sum of distinct terms from this sequence, we say that the sequence is complete. The general problem is : Characterize complete sequences. This Memorandum shows tthat 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. Consideration of a problem that has received considerable attention recently. Given a sequence of positive integers, if every sufficiently large integer can be represented as a sum of distinct terms from this sequence, we say that the sequence is complete. The general problem is: Characterize complete sequences. This Memorandum 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. 32 pp. Ref. (Author)

This report is part of the RAND Corporation research memorandum series. The Research Memorandum was a product of the RAND Corporation from 1948 to 1973 that represented working papers meant to report current results of RAND research to appropriate audiences.

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