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

Jon H. Folkman

ResearchPublished 1965

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)

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Folkman, Jon H., On the representation of integers as sums of distinct terms from a fixed sequence. Santa Monica, CA: RAND Corporation, 1965. https://www.rand.org/pubs/research_memoranda/RM4528.html.
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