Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.4

The Inverse Problem on Subset Sums, II

Jian-Dong Wu
School of Mathematical Sciences and Institute of Mathematics
Nanjing Normal University
Nanjing 210023
P. R. China


For a set T of integers, let P(T) be the set of all finite subset sums of T, and let T(x) be the set of all integers of T not exceeding x. Let $B=\{b_{1}<b_{2}<\cdots \}$be a sequence of integers and d1=10, d2=3b1+4, and dn=3bn-1+2 $(n\ge 3)$. In this paper, we prove that

(i) if bn>dn for all $n\ge 1$, then there exists a sequence of positive integers $A=\{a_{1}<a_{2}<\cdots \}$such that, for all $k\ge 2$, $P(A(b_k))=[0,2b_k]\setminus \{b_u,
2b_k-b_u : 1\le u\le k\}$;

(ii) if bm=dm for some $m\ge 1$ and bn>dn for all $n\not= m$, then there is no such sequence A.

We also pose a problem for further research.

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Received May 7 2013; revised version received September 5 2013. Published in Journal of Integer Sequences, October 12 2013.

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