Smarandache函数的一些推广
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摘要
任意给定素数p以及正整数α,利用初等的方法和技巧,依据α的p进制表达式的性质给出Smarandache函数S(p~α)的准确计算公式,由此得到任意正整数n的Smarandache函数S(n)的准确计算公式.最后给出Smarandache函数的几类推广函数及其性质.
Let p be a prime and α be a positive integer. Based on elementary methods and techniques,the explicit formula for the Smarandache function S( p~α) is obtained depending on the properties of p-adic expression of α. Furthermore,for any positive integer n the explicit formula of the Smarandache function S( n) is given. Finally,several generalized Smarandache functions are defined and some properties are discussed.
Let p be a prime and α be a positive integer. Based on elementary methods and techniques,the explicit formula for the Smarandache function S( p~α) is obtained depending on the properties of p-adic expression of α. Furthermore,for any positive integer n the explicit formula of the Smarandache function S( n) is given. Finally,several generalized Smarandache functions are defined and some properties are discussed.
引文
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[3]KASHIHARA K.Comments and Topics on Smarandache Notions and Problems[D].New Mexica:Erhus University,1996.
[4]LE M H.A lower bound for(2p-1(2p-1))[J].Smarandache Notions J,2001,12(1):217-218.
[5]廖群英,罗文力.Smarandache函数的准确计算公式以及相关数论方程的求解[J].四川师范大学学报(自然科学版),2017,40(1):1-10.
[6]LIU Y M.On the solutions of an equation involving the Smarandache function[J].Scientia Magna,2006,2(1):76-79.
[7]SMARANDACHE F.Only Problems,Not Solution[M].Chicago:Xiquan Publishing House,1993.
[8]WEN T D.A lower bound estimate of the Smarandache function[J].Pure and Applied Mathematics,2010,26(3):413-416.
[9]XU Z F.The value distribution of Smarandache function[J].Acta Mathematica Sinica,2006,49(5):1009-1012.
[10]XING W Y.On the Smarandache function[C]//Research on Smarandache Problem in Number Theory,2005,2:103-106.
[11]YI Y.An equation in volving the Euler function and Smarandache function[J].Scientia Magna,2005,1(2):172-175.
[12]张文鹏.关于Smarandache函数的两个问题[J].西北大学学报(自然科学版),2008,38(2):173-176.
[2]GORSKI D.The pseudo-smarandache function[J].Smarandache Notions J,2002,13(1/2/3):140-149.
[3]KASHIHARA K.Comments and Topics on Smarandache Notions and Problems[D].New Mexica:Erhus University,1996.
[4]LE M H.A lower bound for(2p-1(2p-1))[J].Smarandache Notions J,2001,12(1):217-218.
[5]廖群英,罗文力.Smarandache函数的准确计算公式以及相关数论方程的求解[J].四川师范大学学报(自然科学版),2017,40(1):1-10.
[6]LIU Y M.On the solutions of an equation involving the Smarandache function[J].Scientia Magna,2006,2(1):76-79.
[7]SMARANDACHE F.Only Problems,Not Solution[M].Chicago:Xiquan Publishing House,1993.
[8]WEN T D.A lower bound estimate of the Smarandache function[J].Pure and Applied Mathematics,2010,26(3):413-416.
[9]XU Z F.The value distribution of Smarandache function[J].Acta Mathematica Sinica,2006,49(5):1009-1012.
[10]XING W Y.On the Smarandache function[C]//Research on Smarandache Problem in Number Theory,2005,2:103-106.
[11]YI Y.An equation in volving the Euler function and Smarandache function[J].Scientia Magna,2005,1(2):172-175.
[12]张文鹏.关于Smarandache函数的两个问题[J].西北大学学报(自然科学版),2008,38(2):173-176.