包含Bernoulli多项式的正整数的m次方部分之和

The Sum of m^th Power Parts of Continued Positive Integers Containing Bernoulli Polynomials

设m是大于1的正整数,对于正整数n,设fm(n)是不小于a的最大m次方幂,Sm(n)是不小于n的所有正整数的m次方部分之和,文章根据连续正整数的齐次和与Bernoulli多项式之间的关系,主要研究Sm(n)的一般性计算公式及其渐近性.

let m be a positive integer with m>1.For any positive integers a and n,let fm(a)denote the maximum mth power number which is not smaller than a,and let Sm(n)denote the sum of fm(a).According to the relation between the homogeneous sums of continued positive integers and Bernoulli polynomials,a computational formula of Sm(n)and an asymptotic property of Sm(n)are the research of this paper.