自然数数码的加法性质(Ⅰ)

Additional Properties of the Digits of a Natural Number(Ⅰ)

研究了数码等幂和 ,指出了当记 A1( n,m)为 n的数码 m次方之和 ,As+1( n,m) =A1( As( n,m) ,m) ( s≥ 1) ,若 k≥ 2 ,ni+1=A1( ni,m) ,i=1,…… ,k- 1,n1=A1( nk,m) ,则称 n1,n2 ,…… ,nk 为一组 m-可交往循环数 .证明了 3个结论 :( 1)给定 n,m,序列 { As( n,m) }中的数值仅有限个不同 .( 2 )给定 m,两组 m-可交往循环数或者集合相等或者集合不相交 .( 3)给定 m,m-可交往循环数仅有有限组 .给出了求全部 m-可交往循环数的算法 ,并利用计算机获得了 m=3,4 ,5,6时的全部 m-可交往循环数 ,最后 ,还提出了 2个猜想 .

This paper study digit's equal power sum.L et A1 (n,m) denote the sum of each digit's m- th power of the natural number n,As+1 (n,m) =A1 (As(n,m) ,m) (s≥ 1 ) .If k≥ 2 ,ni+1 =A1 (ni,m) ,i=1 ,…… ,k- 1 ,n1 =A1 (nk,m) ,then we call n1 ,n2 ,…… ,nk as a set of m- alternant cycle numbers.This article proves the following three propositions:(1 ) Given n,m, only are there finite many numbers in the sequence{As(n,m) }.(2 ) Two sets of m- alternant cycle numbers are equal or not intersecting.(3) Only are there finite many sets of m- alternant cycle numbers.We give a program of computing all sets of m- alternant cycle numbers and by using computer,all sets of m- alternantcycle numbers for m=3,4,5 ,6 .In the final,we provide with two conjectures.