Article(id=1242120552563938194, tenantId=1146029695717560320, journalId=1146031591421210625, issueId=1242120519928058647, articleNumber=null, orderNo=null, doi=null, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1263744000000, receivedDateStr=2010-01-18, revisedDate=1280073600000, revisedDateStr=2010-07-26, acceptedDate=null, acceptedDateStr=null, onlineDate=1284307200000, onlineDateStr=2010-09-13, pubDate=1284307200000, pubDateStr=2010-09-13, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1284307200000, onlineIssueDateStr=2010-09-13, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774074217212, creator=sys-migrate, updateTime=1774074217212, updator=sys-migrate, issue=Issue{id=1242120519928058647, tenantId=1146029695717560320, journalId=1146031591421210625, year='2010', volume='28', issue='17', pageStart='3', pageEnd='140', issueExtLink='null', onlineDate='null', pubDate='1284307200000', pubDateStr='2010-09-13', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=3, issueType=-1, specialIssue=null, createTime=1774074209431, creator='sys-migrate', updateTime=1774074209431, updator='sys-migrate', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=50, endPage=53, ext={EN=ArticleExt(id=1242120555390894873, articleId=1242120552563938194, tenantId=1146029695717560320, journalId=1146031591421210625, language=EN, title=Some Notes on the Smarandache Power Function, columnId=1242116080374710456, journalTitle=Science & Technology Review, columnName=Articles, runingTitle=null, highlight=null, articleAbstract=For any positive integern, the Smarandache power function SP(n) is defined as the smallest positive integermsuch that mm is
divisible by n. The main purpose of this paper is to study the solvability of equations SP(n)=?准(n), k=1, 2, 3 (for the Euler function) and all the positive integer solutions and other related issues, based on the nature of the series{SP(n)}, 1st mean, asymptotic formula, the
convergence of infinite series SP(n) and its related identity. The analytic methods are used to get the distribution properties of the k-th
Powers of SP(n). For any real number x≥3, given the real numbers k, l (k>0, l≥0), and all the prime numbers p, any positive number ε
and the Riemann Zeta-function, we give and prove the corresponding asymptotic formula. For any real number x≥3 and a given real number k′>0, we also give and prove the corresponding asymptotic formula. For any given real numbers x≥3 and real numbers l≥0, the corresponding asymptotic formula is also given and proven together. Thus, the asymptotic formula of ■nl(SP(n))k and ■■(k>0, l≥0) is given, when l=0, k=1/k′, k=1,2,3 and ζ(2)=π2/6, ζ(4)=π4/90, and it could be found that the theorem is a further extension of the related results., authors=null, authorsList=null, authorCompany=null, correspAuthors=null, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=VmTxD44utv2tCIS99AZwrg==, pdfFileSize=1048520, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, fund=null), CN=ArticleExt(id=1242120554631725839, articleId=1242120552563938194, tenantId=1146029695717560320, journalId=1146031591421210625, language=CN, title=关于Smarandache幂函数的注记, columnId=1146540929516700224, journalTitle=科技导报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=对于正整数n，Smarandache幂函数SP(n)定义为最小的正整数m使得n整除mm。本文在研究数列{SP(n)}性质的基础上，通过对SP(n)的一次均值及其渐近公式、无穷数列SP(n)的收敛性及其相关的恒等式、方程SP(nk)=φ(n)(k=1, 2, 3)的可解性（φ(n)为Euler函数）及其所有的正整数解等相关问题的讨论，应用解析方法研究了SP(n)的k次方幂的分布性质。针对任意的实数x≥3、给定的实数k，l（k>0，l≥0），及对所有的素数p、任意的正数ε和Riemann Zeta-函数，给出并证明了其相应的渐近公式；对于任意的实数x≥3及给定的实数k′>0的情况，也给出并证明了其相应的渐近公式；对于任意的实数x≥3及给定的实数l≥0，其相应的渐近公式也一并给出并加以证明。由此，给出■nl(SP(n))k及■■（k>0，l≥0）的渐近公式。在l=0，k=1/k′情况下，以及k=1, 2, 3且ζ(2)=π2/6，
ζ(4)=π4/90情况下，可以看出该定理是对相关结论的进一步推广。, authors=^{任 鹏1，王 阳2，邓书显1}, authorsList=任 鹏;王 阳;邓书显, authorCompany=1． 河南工程学院数理系，郑州 4500072． 南阳师范学院数学与统计学院，河南南阳 473061, correspAuthors=null, authorNote=null, correspAuthorsNote=邓书显, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=l9MZfFWnkVfiTZ2d1dCJqA==, pdfFileSize=1048520, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, fund=null)}, authors=null, keywords=[Keyword(id=1242120554354901770, tenantId=1146029695717560320, journalId=1146031591421210625, articleId=1242120552563938194, language=CN, orderNo=1, keyword=幂函数), Keyword(id=1242120554442982155, tenantId=1146029695717560320, journalId=1146031591421210625, articleId=1242120552563938194, language=CN, orderNo=1, keyword=k次方幂), 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