Smarandache function

The Smarandache function S:++ 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 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 numbersMathworldPlanetmath 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, are given by Sloane’s ( OEIS njas/sequences/?q=A002034A002034.

A list of sixteen s denoted s1 to s16 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 ( OEIS njas/sequences/?q=A048799A048799) is defined as s1=n=2(S(n)!)-11.09317

The second Smarandache constant s2 (Sloane’s ( OEIS njas/sequences/?q=A048834A048834) is defined as s2=n=2S(n)n!1.71400629359162 and it is an irrational number.

The third Smarandache constant s3 (Sloane’s ( OEIS njas/sequences/?q=A048835A048835) is defined as s3=n=2(i=2nS(i))-10.719960700043.

The series s4(α)=n=2nα(i=2nS(i))-1 convergesPlanetmathPlanetmath for a fixed real number α1. Since s4 is a function of α it is not a single constant, but an infinite list of them. The values for small α have been computed:

s4(1)1.72875760530223 (Sloane’s ( OEIS njas/sequences/?q=A048836A048836).

s4(2)4.50251200619296 (Sloane’s ( OEIS njas/sequences/?q=A048837A048837).

s4(3)13.0111441949445 (Sloane’s ( OEIS njas/sequences/?q=A048838A048838).

The fifth Smarandache constant s5=n=1(-1)n-1S(n)n! converges to an irrational number.

Burton in 1995 showed that the series s6=n=2S(n)(n+1)! converges and is boundedPlanetmathPlanetmathPlanetmathPlanetmath by 0.218282<s6<0.5.

Dumitrescu and Seleacu in 1996 showed that the series s7(r)=n=rS(n)(n+r)! and s8(r)=n=rS(n)(n-r)! converge for r+.

The same authors show that the series s9=n=2(i=2nS(i)i!)-1 is convergentMathworldPlanetmath.

The series s10(α)=n=2(S(n))-α(S(n)!)-12 and s11(α)=n=2(S(n))-α[(S(n)-1)!]-12 converge for α>1. These two series also define an infinite list of constants.

If f: is a function satisfying the condition f(t)ctαd(t!)-d((t-1)!), where t is a positive integer, d denotes the divisor functionMathworldPlanetmath, and the given constants α>1, c>1, then the series s12(f)=n=1f(S(n)) is convergent.

The series s13=n=1(k=1nS(k)!)-1n is convergent.

The series s14(α)=n=1(S(n)!)-32(logS(n))-α is convergent for α>1.

The series s15=n=12nS(2n)! is convergent.

The series s16(α)=n=1S(n)n1+α is convergent for α>1.

1. Dumitrescu C, Popescu M, Seleacu V, Tilton H (1996). The Smarandache Function in Number TheoryMathworldPlanetmathPlanetmath. 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).”Calculating Smarandache function in parallel”. Parallel and Distributed Computing, 2004. Third International Symposium on AlgorithmsMathworldPlanetmath, Models and Tools for Parallel Computing on Heterogeneous Networks,: pp.79-82.

4. Kempner AJ (1918). ”Miscellanea”. The American Mathematical Monthly 25: 201-210.

5. Mehendale DP (2005). Classical Smarandache Function and a Formula for Twin PrimesMathworldPlanetmath.

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

7. Smarandache F. smarandache/CONSTANT.TXTConstants Involving the Smarandache Function.

8. Muller R (1990). ”Editorial”. smarandache/SFJ1.pdfSmarandache Function Journal 1: 1.

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

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

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

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

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

14. Burton E (1996). ”On Some Convergent Series”. 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