李海龍,周軍

(渭南師范學(xué)院數(shù)學(xué)與應(yīng)用數(shù)學(xué)研究所,陜西渭南 714000)

關(guān)于位數(shù)函數(shù)高次均值的計算

李海龍,周軍

(渭南師范學(xué)院數(shù)學(xué)與應(yīng)用數(shù)學(xué)研究所,陜西渭南 714000)

針對1993年美國數(shù)論專家Smarandache提出了初等數(shù)論及集合論中的105個未解決的問題中的5個關(guān)于自然數(shù)列的性質(zhì)問題,就自然數(shù)列的位數(shù)函數(shù)問題進(jìn)行了研究,給出了在一個正整數(shù)的n進(jìn)制表示中的位數(shù)函數(shù)定義,采用了歸納、猜想的方法得出了位數(shù)函數(shù)a(m,n)的高次均值的精確計算公式.

n進(jìn)制;位數(shù)函數(shù);均值

1 引言及結(jié)論

在文[1]中美國數(shù)論專家Smarandache提出了初等數(shù)論及集合論中105個未解決的問題讓大家研究,其中有五個問題是關(guān)于自然數(shù)列的性質(zhì)問題.文[2-4]給出了在一個正整數(shù)的n進(jìn)制表示中數(shù)字之和函數(shù)以及它的一次、二次、三次均值的精確計算公式,文[3]主要研究數(shù)字之積函數(shù)的均值,文[5]給出了位數(shù)函數(shù)三次均值的計算公式.本文主要用了歸納、猜想的方法研究自然數(shù)列的位數(shù)函數(shù),得出了在一個正整數(shù)n進(jìn)制表示中的位數(shù)函數(shù)a(m,n)均值A(chǔ)r(m,n),(r=1,2,3,···)的精確計算公式,并給出了證明.

2 引理及其證明

3 定理的證明

參考文獻(xiàn)

[1]Smarandche Florentin.Only Problems,Not Solutions![M].Chicago:Xiquan Publishing House,1993.

[2]李海龍,楊倩麗.關(guān)于n進(jìn)制及其有關(guān)計數(shù)函數(shù)[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2002,18(3):13-15.

[3]楊倩麗,李海龍.關(guān)于n進(jìn)制中數(shù)字之和函數(shù)均值的計算[J].西北大學(xué)學(xué)報,2002,32(4):361-362.

[4]李海龍,楊倩麗.n進(jìn)制中非零數(shù)字之積函數(shù)的均值公式[J].數(shù)學(xué)的實踐與認(rèn)識,2002,32(4):683-686.

[5]楊倩麗,仇英利,李海龍.關(guān)于位數(shù)函數(shù)三次均值的計算[J/OL].西北大學(xué)學(xué)報:自然科學(xué)網(wǎng)絡(luò)版,2006,4(4):0218 [2006-07-10].http://jonline.nwu.edu.cn/wenzhang/206171.pdf.

On high power mean values computations of the location number function

LI Hai-long,ZHOU Jun
(Institute of Mathematics and Applied Mathematics,Weinan Teachers University, Weinan714000,China)

This article was inferred to 105 unsolution questions in primary theory of numbers and set theory which were proposed by American theory of numbers expert Smarandache in 1993 about the nature question of natural sequence,conducted the figure function question of the natural sequence.It was produced n the figure function definition in a positive integer in the system expression,and has used induction suspected methods then has obtained two direct the principle which was related to,also using these directed the principle to obtain Figure function a(m,n);High power mean values as mean as the precise formula on.

base n,function of the location number,average value

O156.4

A

1008-5513(2009)04-0770-04

2007-10-10.

陜西省自然科學(xué)基金(SJ08A22).

李海龍(1963-),教授,研究方向：數(shù)論.

2000MSC:11B83