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In number theory, the Smarandache function S(n) (sequence A002034 in OEIS) is defined for a given positive integer n to be the smallest number such that n divides its factorial.[1][2][3][4][5] For example, the number 8 does not divide 1!, 2!, 3!, but does divide 4!, so S(8)=4.

Historically, the function was first considered by François Édouard Anatole Lucas in 1883,[6] followed by Joseph Jean Baptiste Neuberg in 1887[7] and A. J. Kempner, who in 1918 gave the first correct algorithm for computing S(n).[8] The function was subsequently rediscovered and named after himself by Florentin Smarandache in 1980.[9][10]

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.[11] 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.

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).[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]
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:

$$\sum_{n=2}^\infty 1/ [S(n)]!=1.09317...$$ (sequence A048799 in OEIS).
$$\sum_{n=2}^{\infty}S(n)/n!\approx 1.71400629359162...$$ (sequence A048834 in OEIS) and is irrational.
$$\sum_{n=2}^{\infty}1/\prod_{i=2}^{n}S(i)\approx 0.719960700043...$$(sequence A048835 in OEIS).
$$\sum_n S(n)^{-\alpha} {S(n)!}^{-1/2} <\infty\, (\alpha>1).$$
$$\sum_n {Z(n)}^{-\alpha} < \infty \,(\alpha > 1).$$

References and notes

^ C. Dumitrescu, M. Popescu, V. Seleacu, H. Tilton (1996). The Smarandache Function in Number Theory. Erhus University Press. ISBN 1-879585-47-2.
^ C. Ashbacher, M.Popescu (1995). An Introduction to the Smarandache Function. Erhus University Press. ISBN 1-879585-49-9.
^ 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.
^ Weisstein, Eric W., "Smarandache Constants" from MathWorld.
^ "Constants Involving the Smarandache Function".
^ Lucas, E. (1883). "Question Nr. 288". Mathesis 3: 232.
^ Neuberg, J. (1887). "Solutions de questions proposées, Question Nr. 288". Mathesis 7: 68–69.
^ Kempner, A. J. (1918). "Miscellanea". American Mathematical Monthly 25 (5): 201–210. doi:10.2307/2972639. JSTOR 2972639.
^ F. Smarandache (1980). "A Function in Number Theory". An. Univ. Timisoara, Ser. St. Mat. 18: 79–88. arXiv:math/0405143. MR 0619740.
^ Jonathan Sondow and Eric Weisstein (2006) "Smarandache Function" at MathWorld.
^ R. Muller (1990). "Editorial". Smarandache Function Journal 1: 1. ISBN 84-252-1918-3.
^ Problem 6674 [1991 ,965], American Mathematical Monthly, 101 (1994), 179.
^ K. Kashihara, "Comments and Topics on Smarandache Notions and Problems." Vail: Erhus University Press, 1996.
^ Pinch (2005). "Some properties of the pseudo-Smarandache function". arXiv:math/0504118 [math.NT].
^ I.Cojocaru, S. Cojocaru (1996). "The First Constant of Smarandache". Smarandache Notions Journal 7: 116–118.
^ I. Cojocaru, S. Cojocaru (1996). "The Second Constant of Smarandache". Smarandache Notions Journal 7: 119–120.
^ I. Cojocaru, S. Cojocaru (1996). "The Third and Fourth Constants of Smarandache". Smarandache Notions Journal 7: 121–126.
^ E. Burton (1995). "On Some Series Involving the Smarandache Function". Smarandache Function Journal 6: 13–15.