On a Result of Lemke and Kleitman

Abstract

Lemke and Kleitman [2] showed that, given a positive integer d and d (necessarily non-distinct) divisors of da₁, …, a_d there exists a subset Q ⊆ {1, …, d} such that d = [sum ]_i∈Qa_i 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, …, a_d 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.