Smarandache function

Definitions

The Smarandache function "S"("n") in number theory is defined for a given positive integer "n" as the smallest positive integer "S(n)" such that its factorial "S"("n")! is divisible by "n".cite book
author = C. Dumitrescu, M. Popescu, V. Seleacu, H. Tilton
title = The Smarandache Function in Number Theory
publisher = Erhus University Press
year = 1996
isbn = 1879585472 ] cite book
author = C. Ashbacher, M.Popescu
title = An Introduction to the Smarandache Function
publisher = Erhus University Press
year = 1995
isbn = 1879585499

] cite journal
doi = 10.1109/ISPDC.2004.15
author = S. Tabirca, T. Tabirca, K. Reynolds, L.T. Yang
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,
pages = pp.79–82
date = 2004] MathWorld|title=Smarandache Constants|urlname=SmarandacheConstants] cite web
url = http://www.gallup.unm.edu/~smarandache/CONSTANT.TXT
title = Constants Involving the Smarandache Function]

For example, the number 8 does not divide 1!, 2!, 3!, but does divide 4!, so that "S"(8)=4.

Historically, the function was first considered by Lucas in 1883,cite journal
url =
author = E. Lucas
title = Question Nr. 288
journal = Mathesis
volume = 3
pages = 232
year = 1883] followed by Neuberg in 1887cite journal
url =
author = J. Neuberg
title = Solutions de questions proposées, Question Nr. 288
journal = Mathesis
volume = 7
pages = 68–69
year = 1887] and Kempner in 1918.cite journal
url =
author = A.J. Kempner
title = Miscellanea
journal = Amer. Math. Monthly
volume = 25
pages = 201–210
year = 1918
doi = 10.2307/2972639] It was subsequently rediscovered by Florentin Smarandache in 1980.cite journal
url = http://arxiv.org/abs/math/0405143
author = F. Smarandache
title = A Function in Number Theory
journal = An. Univ. Timisoara, Ser. St. Mat.
volume = 18
pages = 79–88
year = 1980] [Jonathan Sondow and Eric Weisstein (2006) [http://mathworld.wolfram.com/SmarandacheFunction.html "Smarandache Function"] at MathWorld.]

The study of "S"("n") is closely related to that of prime numbers because a number "p" greater than 4 is prime if and only if $S(p)=p$. [cite journal
author = R. Muller
title = Editorial
journal = Smarandache Function Journal
url = http://www.gallup.unm.edu/~smarandache/SFJ1.pdf
volume = 1
issue =
pages = 1
year = 1990]

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 withthe largest prime factor of "n" for "almost all" "n" (in the sense that the asymptotic density of the set of exceptions is zero). [Problem 6674 [1991 ,965] , American Mathematical Monthly, 101 (1994), 179.]

The pseudo-Smarandache function $Z(n)$ is defined for a given "n" as the smallest positive integer "Z(n)" such that "Z"("n")•("Z"("n")+1)/2 is divisible by "n". [K. Kashihara, "Comments and Topics on Smarandache Notions and Problems." Vail: Erhus University Press, 1996.] [ R.G.E. Pinch, " [http://arXiv:math/0504118 Some properties of the pseudo-Smarandache function] "(2005), the only Smarandache-related article to be classified as Number Theory on the Cornell arxiv.]

Associated series

Various series constructed from $S(n)$ and $Z(n)$ have been shown to be convergent. [cite journal
author = I.Cojocaru, S. Cojocaru
title = The First Constant of Smarandache
journal = Smarandache Notions Journal
url = http://www.gallup.unm.edu/~smarandache/SNJ7.pdf
volume = 7
pages = 116–118
year = 1996] [cite journal
author = I. Cojocaru, S. Cojocaru
title = The Second Constant of Smarandache
journal = Smarandache Notions Journal
url = http://www.gallup.unm.edu/~smarandache/SNJ7.pdf
volume = 7
pages = 119–120
year = 1996] [cite journal
author = I. Cojocaru, S. Cojocaru
title = The Third and Fourth Constants of Smarandache
journal = Smarandache Notions Journal
url = http://www.gallup.unm.edu/~smarandache/SNJ7.pdf
volume = 7
pages = 121–126
year = 1996] [ cite journal
author = E. Burton
title = On Some Series Involving the Smarandache Function
journal = Smarandache Function Journal
url = http://www.gallup.unm.edu/~smarandache/SFJ6.pdf
volume = 6
issue =
pages = 13–15
year = 1995] 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 followingare examples of such series:

*$sum\_\{n=2\}^infty\; 1/\; [S(n)]\; !=1.09317...$ OEIS|id=A048799.

*$sum\_\{n=2\}^\{infty\}S(n)/n!approx\; 1.71400629359162...$ OEIS|id=A048834 and is irrational.

*$sum\_\{n=2\}^\{infty\}1/prod\_\{i=2\}^\{n\}S(i)approx\; 0.719960700043...$ OEIS|id=A048835.

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References and notes

ee also

*Florentin Smarandache

External links

* [http://www.geocities.com/m_l_perez/ScientiaMagna.htm Scientia Magna] , a journal so far devoted solely to the Smarandache function, edited in China and available online at Smarandache's website [http://www.gallup.unm.edu/~smarandache/ScientiaMagna1no1.pdf 1-1] [http://www.gallup.unm.edu/~smarandache/ScientiaMagna1no2.pdf 1-2] , [http://www.gallup.unm.edu/~smarandache/ScientiaMagna2no1.pdf 2-1] , [http://www.gallup.unm.edu/~smarandache/ScientiaMagna2no2.pdf 2-2] .