素数幂最大公因数序列和的性质

Properties on the Sum of Sequence for the Greatest Common Divisor of Prime Power

关键词:  素数  最大公因数序列  发散

Key words:  prime  greatest common factor sequence  divergence

摘要:

给出素数幂的最大公因数序列和S(n)=∑_k=1ⁿ d(k),S_α(n)=∑_k=1ⁿ d^α(k)的具体公式,其中,p为素数,k为正整数,d(k)=gcd(p^k+1,p^pk-1+1),并证明S_α(n)(n→∞)是发散的.

Abstract:

Let p be a prime number, let k be a positive integer, and d(k)=gcd (p^k+1,p^pk-1+1).The particular formula of the sum of sequence of the greatest common divisor of prime power was given to be as S(n)=∑_k=1ⁿ d(k),S_α(n)=∑_k=1ⁿ d^α(k), and the scatter of S_α(n)(n→∞)was proved.