A Few Questions About Sequences Of Integers

In this thesis,we study three problems about sequences,the upper bound of density of sets without k terms geometric progressions,inequalities of divisor function on arithmetic progressions,sum and generalization of double recursive Fibonacci numbers.In the first part,we construct a series of geometric progressions disjoint each other and improve the upper bound for the density of positive integer sequences that contain no k-term geometric progression.Let GPFk(n)be the set of all subsets of {1,2,…,n} which contain no k-term geometric progression,and let Du(A)be the upper asymptotic density of the set A.We get a better conclusion of Du(Gk),if k?3,n?2k-1 and Gk? GPFk(n).In the second part,we prove some upper and lower bounds for the sum of divisor functions.Specifically,this part mainly gives some upper and lower bounds for the partial sum ?i=0m(-1)id(6i+1),?i=0m(-1)id(6i+5)and ?i=0md(6i+1)-?i=0md(6i+5),where d(n)is divisor function.In the last part,we supplement the results of Chaves,and then make similar study on the linear recurrence sequences of order two under generalized conditions.We get two partial sums of symmetric function H(m,n)under the special initial conditions.The sum of all H(m,n)on the main diagonal and below,?i,j=1,j?imH(i,j),and the sum of all H(m,n)on the square matrix m×m,?i,j=1mH(i,j).