Smarandache function

Smarandache function

The Smarandache function $S:{\mathbb{Z}}^{+}\to {\mathbb{Z}}^{+}$ is defined as follows: $S(n)$ is the smallest integer such that $S(n)!$ is divisible by $n$. For example, the number 8 does not divide $1!$, $2!$, $3!$, but does divide $4!$. Therefore $S(8)=4$. Another study of $S(n)$ has been published by http://genealogy.math.ndsu.nodak.edu/html/id.phtml?id=12376Aubrey J. Kempner in 1918, and later the function $S(n)$ has been rediscovered and studied by Florentin Smarandache in 1980. A profound study of this function would contribute to the study of prime numbers in accordance with the following property: if $p$ is a number greater than $4$, then $p$ is a prime if and only if $S(p)=p$. The values of $S(n)$ for $n=1,2,3,\mathrm{\dots }$ are given by Sloane’s (http://planetmath.org/NeilSloane) OEIS http://www.research.att.com/ njas/sequences/?q=A002034A002034.

A list of sixteen s denoted $s_1$ to $s_{16}$ have been defined with the use of the Smarandache function $S(n)$, and they should not be confused with the Smarandache constant, which is the smallest solution to the generalized Andrica conjecture.

The first Smarandache constant (Sloane’s (http://planetmath.org/NeilSloane) OEIS http://www.research.att.com/ njas/sequences/?q=A048799A048799) is defined as $s_1={\displaystyle \sum _{n=2}^{\mathrm{\infty }}}{\left(S(n)!\right)}^{-1}\approx 1.09317\mathrm{\dots }$

The second Smarandache constant $s_2$ (Sloane’s (http://planetmath.org/NeilSloane) OEIS http://www.research.att.com/ njas/sequences/?q=A048834A048834) is defined as $s_2={\displaystyle \sum _{n=2}^{\mathrm{\infty }}}{\displaystyle \frac{S(n)}{n!}}\approx 1.71400629359162\mathrm{\dots }$ and it is an irrational number.

The third Smarandache constant $s_3$ (Sloane’s (http://planetmath.org/NeilSloane) OEIS http://www.research.att.com/ njas/sequences/?q=A048835A048835) is defined as $s_3={\displaystyle \sum _{n=2}^{\mathrm{\infty }}}{\left({\displaystyle \prod _{i=2}^n}S(i)\right)}^{-1}\approx 0.719960700043\mathrm{\dots }$.

The series $s_4(\alpha )={\displaystyle \sum _{n=2}^{\mathrm{\infty }}}{n}^{\alpha }{\left({\displaystyle \prod _{i=2}^n}S(i)\right)}^{-1}$ converges for a fixed real number $\alpha \ge 1$. Since $s_4$ is a function of $\alpha$ it is not a single constant, but an infinite list of them. The values for small $\alpha$ have been computed:

$s_4(1)\approx 1.72875760530223\mathrm{\dots }$ (Sloane’s (http://planetmath.org/NeilSloane) OEIS http://www.research.att.com/ njas/sequences/?q=A048836A048836).

$s_4(2)\approx 4.50251200619296\mathrm{\dots }$ (Sloane’s (http://planetmath.org/NeilSloane) OEIS http://www.research.att.com/ njas/sequences/?q=A048837A048837).

$s_4(3)\approx 13.0111441949445\mathrm{\dots }$ (Sloane’s (http://planetmath.org/NeilSloane) OEIS http://www.research.att.com/ njas/sequences/?q=A048838A048838).

The fifth Smarandache constant $s_5={\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^{n-1}S(n)}{n!}}$ converges to an irrational number.

Burton in 1995 showed that the series $s_6={\displaystyle \sum _{n=2}^{\mathrm{\infty }}}{\displaystyle \frac{S(n)}{(n+1)!}}$ converges and is bounded by .

Dumitrescu and Seleacu in 1996 showed that the series $s_7(r)={\displaystyle \sum _{n=r}^{\mathrm{\infty }}}{\displaystyle \frac{S(n)}{(n+r)!}}$ and $s_8(r)={\displaystyle \sum _{n=r}^{\mathrm{\infty }}}{\displaystyle \frac{S(n)}{(n-r)!}}$ converge for $r\in {\mathbb{Z}}^{+}$.

The same authors show that the series $s_9={\displaystyle \sum _{n=2}^{\mathrm{\infty }}}{\left({\displaystyle \sum _{i=2}^n}{\displaystyle \frac{S(i)}{i!}}\right)}^{-1}$ is convergent.

The series $s_{10}(\alpha )={\displaystyle \sum _{n=2}^{\mathrm{\infty }}}{\left(S(n)\right)}^{-\alpha }{\left(S(n)!\right)}^{-\frac{1}{2}}$ and $s_{11}(\alpha )={\displaystyle \sum _{n=2}^{\mathrm{\infty }}}{\left(S(n)\right)}^{-\alpha }{\left[\left(S(n)-1\right)!\right]}^{-\frac{1}{2}}$ converge for $\alpha >1$. These two series also define an infinite list of constants.

If $f:\mathbb{N}\to ℝ$ is a function satisfying the condition $f(t)\le {\displaystyle \frac{c}{t^{\alpha }d\left(t!\right)-d\left((t-1)!\right)}}$, where $t$ is a positive integer, $d$ denotes the divisor function, and the given constants $\alpha >1$, $c>1$, then the series $s_{12}(f)={\displaystyle \sum _{n=1}^{\mathrm{\infty }}}f\left(S(n)\right)$ is convergent.

The series $s_{13}={\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\left({\displaystyle \prod _{k=1}^n}S(k)!\right)}^{-\frac{1}{n}}$ is convergent.

The series $s_{14}(\alpha )={\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\left(S(n)!\right)}^{-\frac{3}{2}}{\left(\log S(n)\right)}^{-\alpha }$ is convergent for $\alpha >1$.

The series $s_{15}={\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{2^n}{S(2^n)!}}$ is convergent.

The series $s_{16}(\alpha )={\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{S(n)}{n^{1+\alpha }}}$ is convergent for $\alpha >1$.

References

1. Dumitrescu C, Popescu M, Seleacu V, Tilton H (1996). The Smarandache Function in Number Theory. Erhus University Press. ISBN 1879585472.

2. Ashbacher C, Popescu M (1995). An Introduction to the Smarandache Function. Erhus University Press. ISBN 1879585499.

3. Tabirca S, Tabirca T, Reynolds K, Yang LT (2004). http://dx.doi.org/10.1109/ISPDC.2004.15”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.

4. Kempner AJ (1918). ”Miscellanea”. http://www.jstor.org/view/00029890/di991004/99p1446d/0 The American Mathematical Monthly 25: 201-210.

5. Mehendale DP (2005). http://arxiv.org/abs/math/0502384The Classical Smarandache Function and a Formula for Twin Primes.

6. Smarandache F (1980). ”A Function in Number Theory”. Analele Univ. Timisoara, Ser. St. Math. 43: 79-88.

7. Smarandache F. http://www.gallup.unm.edu/ smarandache/CONSTANT.TXTConstants Involving the Smarandache Function.

8. Muller R (1990). ”Editorial”. http://www.gallup.unm.edu/ smarandache/SFJ1.pdfSmarandache Function Journal 1: 1.

9. Cojocaru I, Cojocaru S (1996). ”The First Constant of Smarandache”. http://www.gallup.unm.edu/ smarandache/SNJ7.pdfSmarandache Notions Journal 7: 116-118.

10. Cojocaru I, Cojocaru S (1996). ”The Second Constant of Smarandache”. http://www.gallup.unm.edu/ smarandache/SNJ7.pdfSmarandache Notions Journal 7: 119-120.

11. Cojocaru I, Cojocaru S (1996). ”The Third and Fourth Constants of Smarandache”. http://www.gallup.unm.edu/ smarandache/SNJ7.pdfSmarandache Notions Journal 7: 121-126.

12. Sandor J (1997). ”On The Irrationality Of Certain Alternative Smarandache Series”. http://www.gallup.unm.edu/ smarandache/SNJ8.pdfSmarandache Notions Journal 8: 143-144.

13. Burton E (1995). ”On Some Series Involving the Smarandache Function”. http://www.gallup.unm.edu/ smarandache/SFJ6.pdfSmarandache Function Journal 6: 13-15.

14. Burton E (1996). ”On Some Convergent Series”. http://www.gallup.unm.edu/ smarandache/SNJ7.pdfSmarandache Notions Journal 7 (1-3): 7-9.

Title	Smarandache function
Canonical name	SmarandacheFunction
Date of creation	2013-03-22 17:04:15
Last modified on	2013-03-22 17:04:15
Owner	dankomed (17058)
Last modified by	dankomed (17058)
Numerical id	47
Author	dankomed (17058)
Entry type	Definition
Classification	msc 11A41
Related topic	GeneralizedAndricaConjecture