## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  866.11017
Autor:  Erdös, Paul; Lewin, Mordechai
Title:  d-complete sequences of integers. (In English)
Source:  Math. Comput. 65, No.214, 837-840 (1996).
Review:  Let A = {a1 < a2 < ...} be an infinite sequence of integers. A is said to be complete if every sufficiently large integer is the sum of distinct elements of A. If every large integer is the sum of ai such that no one divides the other, then A is called d-complete.
In 1959, B. J. Birch [Proc. Camb. Philos. Soc. 55, 370-373 (1959; Zbl 093.05003)] proved that the set {p\alpha q\beta | (p,q) = 1; \alpha,\beta in N} is complete. The main result of the paper is the following: The sequences

A1 = {2\alpha 5\beta p\gamma | \alpha,\beta,\gamma in N;  6 < p < 20;  p is prime};    A2 = {3\alpha 5\beta 7\gamma | \alpha,\beta,\gamma in N}

are d-complete. Furthermore, the authors prove: the set

{p\alpha q\beta | p,q > 0;  \alpha,\beta in N}

is d-complete if and only if {p,q} = {2,3}.
Reviewer:  N.Hegyvari (Budapest)
Classif.:  * 11B75 Combinatorial number theory
11B83 Special sequences of integers and polynomials
Keywords:  d-complete sequences of integers
Citations:  Zbl 093.05003

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