Erdős, Paul; Lewin, Mordechai \(d\)-complete sequences of integers. (English) Zbl 0866.11017 Math. Comput. 65, No. 214, 837-840 (1996). 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 \(\{p,q\}= \{2,3\}\). Reviewer: N.Hegyvari (Budapest) Cited in 5 ReviewsCited in 5 Documents MSC: 11B75 Other combinatorial number theory 11B83 Special sequences and polynomials Keywords:\(d\)-complete sequences of integers Citations:Zbl 0093.05003 PDFBibTeX XMLCite \textit{P. Erdős} and \textit{M. Lewin}, Math. Comput. 65, No. 214, 837--840 (1996; Zbl 0866.11017) Full Text: DOI Online Encyclopedia of Integer Sequences: Positive integers which cannot be written as a sum of distinct numbers of the form 4^a + 5^b (a,b >= 0). 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. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.