On the reciprocal sum of a sum-free sequence

Let A = {1 a (c) 3/4 a (1) < a (2) < aEuro broken vertical bar} be a sequence of integers. A is called a sum-free sequence if no a (i) is the sum of two or more distinct earlier terms. Let lambda be the supremum of reciprocal sums of sum-free sequences. In 1962, ErdAs proved that lambda < 103. A sum-free sequence must satisfy a (n) a (c) 1/2 (k + 1)(n - a (k) ) for all k, n a (c) 1/2 1. A sequence satisfying this inequality is called a kappa-sequence. In 1977, Levine and O'sullivan proved that a kappa-sequence A with a large reciprocal sum must have a (1) = 1, a (2) = 2, and a (3) = 4. This can be used to prove that lambda < 4. In this paper, it is proved that a kappa-sequence A with a large reciprocal sum must have its initial 16 terms: 1, 2, 4, 6, 9, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, and 50. This together with some new techniques can be used to prove that lambda < 3.0752. Three conjectures are posed.