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**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 = **{**a_{1} < a_{2} < ...**}** 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 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 A_{1} = **{**2^{\alpha} 5^{\beta} p^{\gamma} | \alpha,\beta,\gamma in **N**; 6 < p < 20; p is prime**}**; A_{2} = **{**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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