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\(d\)-complete sequences of integers. (English) Zbl 0866.11017

Let \(A=\{a_1<a_2<\dots\}\) 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 \(a_i\) 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 0093.05003)] proved that the set \(\{p^\alpha q^\beta\mid (p,q)=1\); \(\alpha,\beta\in \mathbb{N}\}\) is complete. The main result of the paper is the following: The sequences \[ A_1=\{2^\alpha 5^\beta p^\gamma\mid \alpha,\beta,\gamma\in \mathbb{N};\;6<p<20;\;p\text{ is prime}\}; \qquad A_2=\{3^\alpha 5^\beta 7^\gamma\mid \alpha,\beta,\gamma\in \mathbb{N}\} \] are \(d\)-complete. Furthermore, the authors prove: the set \[ \{p^\alpha q^\beta\mid p,q>0;\;\alpha,\beta\in \mathbb{N}\} \] is \(d\)-complete if and only if \(\{p,q\}= \{2,3\}\).

MSC:

11B75 Other combinatorial number theory
11B83 Special sequences and polynomials

Citations:

Zbl 0093.05003
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References:

[1] B. J. Birch, On 3\? points in a plane, Proc. Cambridge Philos. Soc. 55 (1959), 289 – 293. · Zbl 0089.38502
[2] J. W. S. Cassels, On the representation of integers as the sums of distinct summands taken from a fixed set, Acta Sci. Math. Szeged 21 (1960), 111 – 124. · Zbl 0217.32102
[3] P. Erdős, Quickie, Math. Mag. 67 (1994), pp. 67 and 74.
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