The Arithmetic Function F(n) And Catalan Numbers Modulo P~k

In this paper,we study the arithmetic function F(n)and Catalan numbers.(a)For any integer n≥2,1et (?)(n)denote the set of all triples (a1,…,as)such that(i)2≤ai≤n are integers for1≤i≤s,(ii)(ai,aj)=1for all1≤i<j≤s,(iii)if a prime p|ai,then p|n.Define F(n)to be the maximum of a1+…+as,where(a1,a2,…, as)∈(?)(n). Define ω(n)to be the number of distinct prime factors of n.In1983,Erdos showed that for any positive integer k,there is an integer nk with F(nk)=nk and ω(nk)=k.In this thesis,we prove the following result:For any positive integer k and a positive number x which is large enough,we have#{0<n≤x:F(n)=n,ω(n)=k}≥(1+o(1))2k-1/Kk/logkx/x(b)Let Cn=(2n)!/((n+1)!n!)be the n-th Catalan number. It is proved that for any odd prime p and integers a,k with0≤a<p and k>0,if0≤a<(p+1)/2,then the Catalan num-bers Cq1-a,…,Cpk-a are all distinct modulo pk,and the sequence (Cpn-a)n≥1modulo pk is constant from n=k on；if(p+1)／2≤a<p, then the Catalan numbers Cp1-a,…,Cpk+1-a are all distinct modulo pk,and the sequence(Cpn-a)n≥1modulo pk is constant from n=k+1on.The similar conclusion is proved for p:2recently by Lin.