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On a Result of Lemke and Kleitman

Published online by Cambridge University Press:  01 March 1997

TRISTAN DENLEY
Affiliation:
Department of Mathematics, Umeå University, S-901 87 Umeå, Sweden

Abstract

Lemke and Kleitman [2] showed that, given a positive integer d and d (necessarily non-distinct) divisors of da1, …, ad there exists a subset Q ⊆ {1, …, d} such that d = [sum ]i∈Qai answering a conjecture of Erdo″s and Lemke. Here we extend this result, showing that, provided [sum ]p|d1/p [les ] 1 (where the sum is taken over all primes p), there is some collection from a1, …, ad which both sum to d and which can themselves be ordered so that each element divides its successor in the order. Furthermore, we shall show that the condition on the prime divisors is in some sense also necessary.

Type
Research Article
Copyright
1997 Cambridge University Press

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