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| [[File:SmarandacheFunction.PNG|thumb|374px|right|Graph of the Kempner function]]
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| In [[number theory]], the '''Kempner function''' ''S''(''n'')<ref>Called the Kempner numbers in the [[Online Encyclopedia of Integer Sequences]]: see {{SloanesRef| A002034 |name=Kempner numbers: smallest number ''m'' such that ''n'' divides ''m''<nowiki>!</nowiki>}}</ref> is defined for a given [[positive integer]] ''n'' to be the smallest number such that ''n'' divides its factorial.
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| The function was first considered by [[François Édouard Anatole Lucas]] in 1883,<ref name="history1">
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| {{cite journal
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| | first = E. | last = Lucas | authorlink = François Édouard Anatole Lucas
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| | title = Question Nr. 288
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| | journal = [[Mathesis (journal)|Mathesis]]
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| | volume = 3
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| | pages = 232
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| | year = 1883
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| }}</ref> followed by [[Joseph Jean Baptiste Neuberg]] in 1887<ref name="history2">
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| {{cite journal
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| | first = J. | last = Neuberg | authorlink = Joseph Jean Baptiste Neuberg
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| | title = Solutions de questions proposées, Question Nr. 288
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| | journal = [[Mathesis (journal)|Mathesis]]
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| | volume = 7
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| | pages = 68–69
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| | year = 1887
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| }}</ref> and [[Aubrey J. Kempner|A. J. Kempner]], who in 1918 gave the first correct algorithm for computing ''S''(''n'').<ref name="history3">
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| {{cite journal
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| | jstor = 2972639
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| | first = A. J. | last = Kempner
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| | title = Miscellanea
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| | journal = [[American Mathematical Monthly]]
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| | volume = 25
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| | pages = 201–210
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| | year = 1918
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| | doi = 10.2307/2972639
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| | issue = 5
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| }}</ref>
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| For example, the number 8 does not divide 1<nowiki>!</nowiki>, 2<nowiki>!</nowiki>, 3<nowiki>!</nowiki>, but does divide 4<nowiki>!</nowiki>, so ''S''(8) = 4.
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| In 1980, following his usual habit, Florentin Smarandache re-named this function after himself.<ref name="history4">
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| {{cite journal
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| | arxiv = math/0405143
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| | author = F. Smarandache
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| | title = A Function in Number Theory
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| | journal = An. Univ. Timisoara, Ser. St. Mat.
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| | volume = 18
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| | pages = 79–88
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| | year = 1980
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| | mr = 0619740
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| }}</ref><ref>Jonathan Sondow and Eric Weisstein (2006) [http://mathworld.wolfram.com/SmarandacheFunction.html "Smarandache Function"] at [[MathWorld]].</ref>
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| Several self-published and possibly pseudonymous books also used the same name,<ref name="smf1">
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| {{cite book
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| | author = C. Dumitrescu, M. Popescu, V. Seleacu, H. Tilton
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| | title = The Smarandache Function in Number Theory
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| | publisher = Erhus University Press
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| | year = 1996
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| | isbn = 1-879585-47-2
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| }}</ref><ref name="smf2">
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| {{cite book
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| | author = C. Ashbacher, M.Popescu
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| | title = An Introduction to the Smarandache Function
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| | publisher = Erhus University Press
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| | year = 1995
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| | isbn = 1-879585-49-9
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| }}</ref>
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| which has now become used more broadly.<ref name="smf3">
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| {{cite journal
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| | doi = 10.1109/ISPDC.2004.15
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| | author = S. Tabirca, T. Tabirca, K. Reynolds, L.T. Yang
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| | title = Calculating Smarandache function in parallel | journal = Parallel and Distributed Computing, 2004. Third International Symposium on Algorithms, Models and Tools for Parallel Computing on Heterogeneous Networks,
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| | pages = pp.79–82
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| | year = 2004
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| }}</ref><ref name="math_world">{{MathWorld|title=Smarandache Constants|urlname=SmarandacheConstants}}</ref><ref name="Smfunctions">
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| {{cite web
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| | url = http://www.gallup.unm.edu/~smarandache/CONSTANT.TXT
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| | title = Constants Involving the Smarandache Function
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| }}</ref>
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| ==Properties==
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| Since ''n'' divides ''n''!, ''S''(''n'') is always at most ''n''. A number ''n'' greater than 4 is a [[prime number]] if and only if ''S''(''n'') = ''n''.<ref>
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| {{cite journal
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| | author = R. Muller
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| | title = Editorial
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| | journal = Smarandache Function Journal
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| | url = http://www.gallup.unm.edu/~smarandache/SFJ1.pdf
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| | volume = 1
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| | issue =
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| | pages = 1
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| | year = 1990
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| | isbn = 84-252-1918-3
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| }}</ref> That is, the numbers ''n'' for which ''S''(''n'') is as large as possible relative to ''n'' are the primes. In the other direction, the numbers for which ''S''(''n'') is as small as possible are the factorials: ''S''(''k''!) = ''k'', for all ''k'' ≥ 1.
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| The Kempner function ''S''(''n'') of an arbitrary number ''n'' is the maximum, over the [[prime power]]s ''p''<sup>''e''</sup> dividing ''n'', of ''S''(''p''<sup>''e''</sup>).<ref name="history3"/>
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| For a number of the form ''n'' = ''px'', where ''p'' is prime and ''x'' is less than ''p'', the Kempner function of ''n'' is ''p''.<ref name="history3"/> It follows from this that computing the Kempner function of a [[semiprime]] (a product of two primes) is computationally equivalent to finding its [[prime factorization]], believed to be a difficult problem.
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| In one of the advanced problems in the ''[[American Mathematical Monthly]]'', set in 1991 and solved in 1994, [[Paul Erdős]] pointed out that the function ''S''(''n'') coincides with the largest [[prime factor]] of ''n'' for "almost all" ''n'' (in the sense that the [[asymptotic density]] of the set of exceptions is zero).<ref>{{citation|title=The smallest factorial that is a multiple of ''n'' (solution to problem 6674)|journal=[[American Mathematical Monthly]]|volume=101|year=1994|page=179|url=http://www-fourier.ujf-grenoble.fr/~marin/une_autre_crypto/articles_et_extraits_livres/irationalite/Erdos_P._Kastanas_I.The_smallest_factorial...-.pdf|first1=Paul|last1=Erdős|author1-link=Paul Erdős|first2=Ilias|last2=Kastanas}}.</ref>
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| ==Associated series==
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| Various series constructed from ''S''(''n'') have been shown to be convergent.<ref>
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| {{cite journal
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| | author = I.Cojocaru, S. Cojocaru
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| | title = The First Constant of Smarandache
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| | journal = Smarandache Notions Journal
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| | url = http://www.gallup.unm.edu/~smarandache/SNJ7.pdf
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| | volume = 7
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| | pages = 116–118
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| | year = 1996
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| }}</ref><ref>
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| {{cite journal
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| | author = I. Cojocaru, S. Cojocaru
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| | title = The Second Constant of Smarandache
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| | journal = Smarandache Notions Journal
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| | url = http://www.gallup.unm.edu/~smarandache/SNJ7.pdf
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| | volume = 7
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| | pages = 119–120
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| | year = 1996
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| }}</ref><ref>
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| {{cite journal
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| | author = I. Cojocaru, S. Cojocaru
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| | title = The Third and Fourth Constants of Smarandache
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| | journal = Smarandache Notions Journal
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| | url = http://www.gallup.unm.edu/~smarandache/SNJ7.pdf
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| | volume = 7
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| | pages = 121–126
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| | year = 1996
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| }}</ref><ref>
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| {{cite journal
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| | author = E. Burton
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| | title = On Some Series Involving the Smarandache Function
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| | journal = Smarandache Function Journal
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| | url = http://www.gallup.unm.edu/~smarandache/SFJ6.pdf
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| | volume = 6
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| | issue =
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| | pages = 13–15
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| | year = 1995
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| }}</ref> In the case of ''S''(''n''), the series have been referred to in the literature as ''Smarandache constants'', even when they depend on auxiliary parameters. Note also that these constants differ from the Smarandache constant that arises in Smarandache's generalization of [[Andrica's conjecture]]. The following are examples of such series:
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| *<math>\sum_{n=2}^\infty 1/ [S(n)]!=1.09317\ldots</math> {{OEIS2C|id=A048799}}.
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| *<math>\sum_{n=2}^{\infty}S(n)/n!\approx 1.71400629359162\ldots</math> {{OEIS2C|id=A048834}} and is [[irrational number|irrational]].
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| *<math>\sum_{n=2}^{\infty}1/\prod_{i=2}^{n}S(i)\approx 0.719960700043\ldots</math> {{OEIS2C|id=A048835}}.
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| *<math> \sum_n S(n)^{-\alpha} {S(n)!}^{-1/2} <\infty\, (\alpha>1). </math>
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| ==References and notes==
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| {{reflist|colwidth=30em}}
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| {{PlanetMath attribution
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| |id = | urlname =SmarandacheFunction
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| |title = Smarandache function}}
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| [[Category:Factorial and binomial topics]]
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