On consecutive sums in sequences

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Let A c N. We define B as the following set of integers, called the set of the consecutive sums in the sequence A: B = { 2 a,[A = {al , as, ... , a,,}, l ~_ u <= v <= n}. u~i~o We will call it shortly the set of c-sums. In [1] Erd6s and Harzheim asked that if 1 <-al, az . . . . . ak<=n, can we find c n a / s so that all c-sums are different? (They conjectured that this is not true if a x < a s < a . . . <ak is also assumed.) We now answer the first question and investigate several related problems. Some authors examined the case of c-sums with just two terms (see Segal, Odlyzko in [1] and see also [2] and [3]). 1. In this section we answer the question of Erd6s and Harzheim. We prove the following TI-mOREM 1. L e t k = f ( n ) be the max imum number o f integers so that 1 <=al, as . . . . . . . . ak ~_n and all c-sums are different. Then