Smarandache function

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The Smarandache function
The Smarandache function

Contents

[edit] 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.[1][2][3][4][5]

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,[6] followed by Neuberg in 1887[7] and Kempner in 1918.[8] It was subsequently rediscovered by Florentin Smarandache in 1980.[9][10]

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.[11]

In one of the advanced problems in the American Mathematical Monthly, set in 1991 and solved in 1994, Paul Erdos 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).[12]

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.[13][14]

[edit] Associated series

Various series constructed from S(n) and Z(n) have been shown to be convergent.[15][16][17][18] 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:

  • \sum_n S(n)^{-\alpha} {S(n)!}^{-1/2} <\infty\, (\alpha>1).
  • \sum_n {Z(n)}^{-\alpha} < \infty \,(\alpha > 1).

[edit] References and notes

  1. ^ C. Dumitrescu, M. Popescu, V. Seleacu, H. Tilton (1996). The Smarandache Function in Number Theory. Erhus University Press. ISBN 1879585472. 
  2. ^ C. Ashbacher, M.Popescu (1995). An Introduction to the Smarandache Function. Erhus University Press. ISBN 1879585499. 
  3. ^ S. Tabirca, T. Tabirca, K. Reynolds, L.T. Yang (2004). "Calculating Smarandache function in parallel". Parallel and Distributed Computing, 2004. Third International Symposium on Algorithms, Models and Tools for Parallel Computing on Heterogeneous Networks,: pp.79–82. doi:10.1109/ISPDC.2004.15. 
  4. ^ Eric W. Weisstein, Smarandache Constants at MathWorld.
  5. ^ Constants Involving the Smarandache Function.
  6. ^ E. Lucas (1883). "Question Nr. 288". Mathesis 3: 232. 
  7. ^ J. Neuberg (1887). "Solutions de questions proposées, Question Nr. 288". Mathesis 7: 68–69. 
  8. ^ A.J. Kempner (1918). "Miscellanea". Amer. Math. Monthly 25: 201–210. doi:10.2307/2972639. 
  9. ^ F. Smarandache (1980). "A Function in Number Theory". An. Univ. Timisoara, Ser. St. Mat. 18: 79–88. 
  10. ^ Jonathan Sondow and Eric Weisstein (2006) "Smarandache Function" at MathWorld.
  11. ^ R. Muller (1990). "Editorial". Smarandache Function Journal 1: 1. 
  12. ^ Problem 6674 [1991 ,965], American Mathematical Monthly, 101 (1994), 179.
  13. ^ K. Kashihara, "Comments and Topics on Smarandache Notions and Problems." Vail: Erhus University Press, 1996.
  14. ^ R.G.E. Pinch, "Some properties of the pseudo-Smarandache function"(2005), the only Smarandache-related article to be classified as Number Theory on the Cornell arxiv.
  15. ^ I.Cojocaru, S. Cojocaru (1996). "The First Constant of Smarandache". Smarandache Notions Journal 7: 116–118. 
  16. ^ I. Cojocaru, S. Cojocaru (1996). "The Second Constant of Smarandache". Smarandache Notions Journal 7: 119–120. 
  17. ^ I. Cojocaru, S. Cojocaru (1996). "The Third and Fourth Constants of Smarandache". Smarandache Notions Journal 7: 121–126. 
  18. ^ E. Burton (1995). "On Some Series Involving the Smarandache Function". Smarandache Function Journal 6: 13–15. 

[edit] See also

[edit] External links

  • Scientia Magna, a journal so far devoted solely to the Smarandache function, edited in China and available online at Smarandache's website 1-1 1-2, 2-1, 2-2.