关于Levine-O'Sullivan序列

Levine-O’Sullivan序列Q={q1,q2,…}定义为q1=1,qn=max1≤k≤n-1{(+1)(n-qk)},n=2,3,…令Wn=(n+1)qn-nqn-1+1,s(n)=1/qn-1/(n(n+1))log(n(Wn+1))/(Wn-n).[Acta Math.Sinica(Chinese Ser.),2015,58(4):529-534]中证明了s(n)>1/n1.667对于所有的n>2500均成立,从而肯定了[Sci.China Math.,2013,56(5):951-966]中提出的猜想.本文证明了limn→∞log s(n)/log n=-√5+1/2.

On Levine–O'Sullivan's Sequence and Its Conjecture

Let ql = 1, and qn be the largest value of （k ＋ 1）（n - qk） for all integers 1 ≤ k≤ n - 1 with n ≥ 2. The sequence Q -- {q1, q2, q3,...} is called Levine-O＇Sullivan sequence. In this paper, we use the combinational and analysis skill and the mathematical induction to study the asymptotic properties of qn, and give an interesting asymptotic formula for it. This solved a conjecture proposed by Professor Chen Yonggao.