Smarandache Functions

Several Smarandache Functions, Prime Numbers, and Constants in number theory are presented below.

The classical Smarandache Function S(n) [] is defined as the smallest integer n such that S(n)! is divisible by n.

Many extensions and versions of this function have been defined and studied in the meantime, such as: Smarandache Ceil Function, Smarandache Kurepa Function, Smarandache Wagstaff Function, Smarandache Near-to-Primordial Function, Pseudo-Smarandache Function, Smarandache Prime, Smarandache-Wellin Number, Smarandache Prime, Smarandache-Wellin Prime, Smarandache Constants.

Eric W. Weisstein, CRC Encyclopedia of Mathematics, CRC Press, Boca Raton, FL, USA, WolframMathWorld:

 Smarandache Function, CRC Encyclopedia of Mathematics, Smarandache Ceil Function, CRC Encyclopedia of Mathematics, Smarandache Kurepa Function, CRC Encyclopedia of Mathematics, Smarandache Wagstaff Function, CRC Encyclopedia of Mathematics, Smarandache Near-to-Primordial Function, CRC Encyclopedia of Mathematics, Pseudo-Smarandache Function, CRC Encyclopedia of Mathematics, Smarandache Number, CRC Encyclopedia of Mathematics, Smarandache-Wellin Number, CRC Encyclopedia of Mathematics, Pictures of Smarandache-Wellin Numbers by David Curran: http://liveatthewitchtrials.blogspot.co.uk/2017/10/fractal-pattern-in-prime-numbers.html Smarandache Prime, CRC Encyclopedia of Mathematics, Smarandache-Wellin Prime, CRC Encyclopedia of Mathematics, Smarandache Constants, CRC Encyclopedia of Mathematics,

BOOKS

Charles Ashbacher: An introduction to the Smarandache Function. Vail: Erhus Univ. Press, 1995, 62 p.

C. Dumitrescu, V. Seleacu: The Smarandache Function. Vail: Erhus Univ. Press, 1996, 136 p.

Sebastian Martin Ruiz: Aplicaciones de la Función de Smarandache y las Funciones Prima y Coprima. Rehoboth: American Research Press, 2002, 26 p.

Sebastian Martin Ruiz: Applications of Smarandache Function, and Prime and Coprime Functions. Rehoboth: American Research Press, 2002, 26 p.

Felice Russo: A Set of New Smarandache Functions, Sequences and Conjectures in Number Theory. Lupton: American Research Press, 2000, 108 p.

Marius Coman: Conjectures on Primes and Fermat Pseudoprimes, Many Based on Smarandache Function (Collected Papers). Columbus: Education Publishing, 2014, 84 p.

Charles Ashbacher: Pluckings from the Tree of Smarandache Sequences and Functions. Hiawatha: American Research Press, 1998, 90 p.

ARTICLES

Ernst Specker, Norbert Hungerbuhler, Micha Wasem: The Ring of Polyfunctions; (on the Smarandache function) 22 p.; arXiv:2106.11788v1 [math.CO] 22 Jun 2021

Tatiana-Corina Dosescu: An Algorithm for Calculating Smarandache's Function and Using Legendre's FormulaSocial Economic Debates, Volume 9, Issue 2,

2020; 10 p.

S. M. S. Islam, A. A. K. Majumdar: Some Results on the Sandor-Smarandache FunctionJ. Sci. Res. 13 (1), 73-84 (2021), 12 p.

Huaning Liu: A survey on Smarandache notions in number theory I: Smarandache function. In Scientia Magna, Vol. 12 (2017), No. 1, 13 pages.

Huaning Liu: A survey on Smarandache notions in number theory II: pseudo-Smarandache function. In Scientia Magna, Vol. 12 (2017), No. 1, 9 pages.

Marius Coman: The MC function and three Smarandache type sequences, diophantine analysis, September 2015, DOI: 10.13140/RG.2.1.1516.3362, 3 pages.

Constantin Dumitrescu. (2010). A Brief History of the "Smarandache Function". 8 p. http://doi.org/10.5281/zenodo.852726.

MaohuaLe. (1998). A Formula of the Smarandache Function. 2 p. http://doi.org/10.5281/zenodo.852732.

Hailong Li. (2008). A generalization of the Smarandache function. In Scientia Magna, Vol. 4 (2008), No. 1, pp. 31-34. http://doi.org/10.5281/zenodo.852736

M. Andrei, I. Balacenoiu, C. Dumitrescu, E. Radescu, & V. Seleacu. (1995). A Linear Combination with Smarandache Function to Obtain the Identity. In Proceedings of Conference of 26th Annual Iranian Math. Conference, 28-31 March 1995, 5 p. http://doi.org/10.5281/zenodo.852754

Henry Ibstedt. (1995). Alternating Iterations of the sum of divisors Functions and the Pseudo-Smarandache Function. In Recreational Mathematics, 13 p. http://doi.org/10.5281/zenodo.852904

Yanchun Guo. (2005). A new additive function and the F. Smarandache function. In Scientia Magna, Vol. 5 (2009), No. 1, pp.128-132. http://doi.org/10.5281/zenodo.852944

Mihaly Bencze. (1997). A New Inequality for the Smarandache Function. 1 p. http://doi.org/10.5281/zenodo.852948

Jing Fu, & Yu Wang. (2008). A new Smarandache function and its elementary properties. In Scientia Magna, Vol. 4 (2008), No. 3, pp. 26-28. http://doi.org/10.5281/zenodo.852950

Henry Ibstedt. (1997). An Illustration of the Distribution of the Smarandache Function. 2 p. http://doi.org/10.5281/zenodo.852952

Maohua Le. (1995). An Inequality Concerning the Smarandache Function. 2 p. http://doi.org/10.5281/zenodo.852954

Weiyi Zhu. (2008). An inequality of the Smarandache function. In Scientia Magna, Vol. 4 (2008), No. 1, pp. 130-133. http://doi.org/10.5281/zenodo.852956

A.A.K. Majumdar. (2008). A note on the near pseudo Smarandache function. In Scientia Magna, Vol. 4 (2008), No. 4, pp. 104-111. http://doi.org/10.5281/zenodo.852960

Sebastian Martin Ruiz. (1998). A result obtained using Smarandache Function. 2 p. http://doi.org/10.5281/zenodo.852964

Henry Ibstedt. (1995). Base Solution (the Smarandache Function). 4 p. http://doi.org/10.5281/zenodo.852966

Mark Farris & Patrick Mitchell. (1997). Bounding the Smarandache Function. 6 pages. http://doi.org/10.5281/zenodo.852972

Sabin Tabirca, Tatiana Tabirca, Kieran Reynolds, & Laurence T Yang. (2004). Calculating Smarandache Function in Parallel. 4 pages. http://doi.org/10.5281/zenodo.852974

J. R. Sutton. (1995). Calculating the Smarandache Function for Powers of a Prime. 3 pages. http://doi.org/10.5281/zenodo.852976

Marius Coman. (2001). Five conjectures on Sophie Germain primes and Smarandache function and the notion of Smarandache-Germain primes. 3 pages. Zenodo. http://doi.org/10.5281/zenodo.852978

Balacenoiu I. & V. Seleacu. (1994). History of The Smarandache Function. 10 pages. http://doi.org/10.5281/zenodo.852990

Yi Yuan, & Zhang Wenpeng. (2013). Mean Value of The Additive Analogue of Smarandache Function. In Scientia Magna Vol. 1, No. 1, pp. 145-147. http://doi.org/10.5281/zenodo.852992

Zhu Minhui. (2013). Mean Value of The Additive Analogue of Smarandache Function. In Scientia Magna Vol. 1, No. 1, pp. 149-182. http://doi.org/10.5281/zenodo.852996

Steven R Finch. (1997). Moments of The Smarandache Function. 2 p. http://doi.org/10.5281/zenodo.852998

A.W. Vyawahare, & K. M. Purohit. (1997). Near Pseudo Smarandache Function. 20 pages.  http://doi.org/10.5281/zenodo.853000

Jozsef Sandor. (1995). On a dual of the Pseudo-Smarandache function. 6 pages. http://doi.org/10.5281/zenodo.853002

J. Sandor. (1995). On an inequality for the Smarandache function. 3 pages. http://doi.org/10.5281/zenodo.853006

J. Sandor. (1997). On certain generalizations of the Smarandache function. 11 pages. http://doi.org/10.5281/zenodo.853010

Sandor Jozsef. (1997). On Certain Inequalities Involving the Smarandache Function. 4 pages. http://doi.org/10.5281/zenodo.853012

Jozsef Sandor. (1995). On certain new inequalities and limits for the Smarandache function. 8 pages. http://doi.org/10.5281/zenodo.853016

Wei Qin. (2008). One problem related to the Smarandache function. In Scientia Magna, Vol. 4 (2008), No. 3, pp. 106-108. http://doi.org/10.5281/zenodo.853018

Charles Ashbacher. (2000). On Numbers where the Values of the Pseudo-Smarandache Function of It and The Reversal Are Identical. 3 pages. http://doi.org/10.5281/zenodo.853022

A.A.K. Majumdar. (2008). On some pseudo Smarandache function related triangles. In Scientia Magna, Vol. 4 (2008), No. 3, pp. 95-105. http://doi.org/10.5281/zenodo.853024

Emil Burton. (2002). On Some Series Involving Smarandache Function. 3 pages. http://doi.org/10.5281/zenodo.853030

C. Dumitrescu, & C. Rocsoreanu. (1995). On the Calculus of Smarandache Function. 8 pages. http://doi.org/10.5281/zenodo.853034

Zhongtian Lv. (2007). On the F.Smarandache function and its mean value. In Scientia Magna, Vol. 3 (2007), No. 2, pp. 104-108. http://doi.org/10.5281/zenodo.853036

Chan Shi. (2010). On the hybrid mean value of the Smarandache kn-digital sequence and Smarandache function. In Scientia Magna, Vol. 6 (2010), No. 4, pp. 20-23. http://doi.org/10.5281/zenodo.853038

J. Sandor. (2011). On the Irrationality of Certain Constants Related to the Smarandache Function. 3 pages. http://doi.org/10.5281/zenodo.853040

Lin Cheng. (2007). On the mean value of the Pseudo-Smarandache function. In Scientia Magna, Vol. 3 (2007), No. 3, pp. 97-100. http://doi.org/10.5281/zenodo.853042

Hai Yang, & Ruiqin Fu. (2006). On the mean value of the Near Pseudo Smarandache Function. In Scientia Magna, Vol. 2 (2006), No. 2, pp. 35-39.  http://doi.org/10.5281/zenodo.853044

Yongfeng Zhang. (2007). On the near pseudo Smarandache function. In Scientia Magna, Vol. 3 (2007), No. 1, pp. 98-101. http://doi.org/10.5281/zenodo.853048

Xuhui Fan. (2008). On the Pseudo-Smarandache-Squarefree function and Smarandache function. In Scientia Magna, Vol. 4 (2008), No. 4, pp. 7-11. http://doi.org/10.5281/zenodo.853050

J. Sandor. (1997). On the Pseudo-Smarandache Function. 3 pages. http://doi.org/10.5281/zenodo.853056

Yani Zheng. (2007). On the Pseudo Smarandache function and its two conjectures. In Scientia Magna, Vol. 3 (2007), No. 4, pp. 74-76. http://doi.org/10.5281/zenodo.853058

Henry Ibstedt. (2011). On the Pseudo-Smarandache Function and Iteration Problems Part II: The Sum of Divisors Function. 5 pages. http://doi.org/10.5281/zenodo.853062

Yuanbing Lou. (2007). On the pseudo Smarandache function. In Scientia Magna, Vol. 3 (2007), No. 4, pp. 48-50. http://doi.org/10.5281/zenodo.853064

Su Gou, & Jianghua Li. (2007). On the Pseudo-Smarandache function. In Scientia Magna, Vol. 3 (2007), No. 4, pp. 81-83. http://doi.org/10.5281/zenodo.853068

Henry Ibstedt. (2011). On the Pseudo-Smarandache Function and Iteration Problems. 8 pages. http://doi.org/10.5281/zenodo.853074

Mingdong Xiao. (2008). On the Smarandache function and the divisor product sequences. In Scientia Magna, Vol. 5 (2009), No. 2, pp. 129-132. http://doi.org/10.5281/zenodo.853076

Jinrui Wang. (2008). On the Smarandache function and the Fermat number. In Scientia Magna, Vol. 4 (2008), No. 2, pp. 25-28. http://doi.org/10.5281/zenodo.853078

Albert A. Mullin. (2000). On the Smarandache Function and The Fixed - Point Theory of Numbers. 1 page. http://doi.org/10.5281/zenodo.853080

Zhang Wenpeng, & Xu Zhefeng. (2004). On the Smarandache Function and Square Complements. In Scientia Magna Vol. 1, No. 1, 3 pages. http://doi.org/10.5281/zenodo.853082

Lu Yaming. (2006). On the solutions of an equation involving the Smarandache function. In Scientia Magna, Vol. 2 (2006), No. 1, pp. 76-79. http://doi.org/10.5281/zenodo.853084

Weiguo Duany, & Yanrong Xue. (2008). On the solvability of an equation involving the Smarandache function and Euler function. In Scientia Magna, Vol. 4 (2008), No. 2, pp. 29-33. http://doi.org/10.5281/zenodo.853086

Mihaly Bencze. (2011). Open Questions for The Smarandache Function. 3 pages. http://doi.org/10.5281/zenodo.853088

Charles Ashbacher. (2011). Palindromic Numbers and Iterations of the Pseudo-Smarandache Function. 2 pages. http://doi.org/10.5281/zenodo.853090

A.S. Muktibodh, & S.T. Rathod. (2011). Pseudo-Smarandache Functions of First and Second Kind. In International J.Math. Combin. Vol. 3, (2011), pp. 17-22 http://doi.org/10.5281/zenodo.853094

Ion Balacenoiu, & Constantin Dumitrescu. (2011). Smarandache Functions of The Second Kind. 4 pages. http://doi.org/10.5281/zenodo.853096

H. Ibstedt. (1993). The Smarandache function S(n). 2 p.  http://doi.org/10.5281/zenodo.853516

Zhong Li. (2004). Solution of Two Questions Concerning the Divisor Function And The Pseudo-Smarandache Function. 4 p.  http://doi.org/10.5281/zenodo.853518

Constantin Dumitrescu, & Cannen Rocsoreanu. (2002). Some Connections Between the Smarandache Function and the Fibonacci Sequence. 11 p.  http://doi.org/10.5281/zenodo.853522

E. Burton, L. Cojocaru, S. Cojocaru, & C. Dumitrescu. (1994). 8 pages. Some Convergence Problems Involving the Smarandache Function. http://doi.org/10.5281/zenodo.853526

I. Radescu, N. Radescu, & C. Dumitrescu. (1992). Some Elementary Algebraic Considerations Inspired by The Smarandache Function. 5 pages. http://doi.org/10.5281/zenodo.853534

Yu Wang. (2007). Some identities involving the near pseudo Smarandache function. In Scientia Magna, Vol. 3 (2007), No. 2, pp. 44-49. http://doi.org/10.5281/zenodo.853540

Kang Xiaoyu. (2005). Some interesting properties of the Smarandache function. In Scientia Magna, Vol. 1 (2005), No. 2, pp. 52-54. http://doi.org/10.5281/zenodo.853596

Yulin Lu. (2007). Some new problems about the Smarandache function and related problems. In Scientia Magna, Vol. 3 (2007), No. 3, pp. 110-111. http://doi.org/10.5281/zenodo.853654

Richard Pinch. (2005). Some properties of the Pseudo-Smarandache function. In Scientia Magna, Vol. 1 (2005), No. 2, pp. 167-172. http://doi.org/10.5281/zenodo.853702

Balacenoiu I. & V. Seleacu. (1990). Some Properties of Smarandache Functions of The Type I. 5 pages. http://doi.org/10.5281/zenodo.853754

Richard Pinch. (2005). Some Properties of The Pseudo-Smarandache Function. In arXiv:math/0504118v1 [math.NT] 6 Apr 2005. http://doi.org/10.5281/zenodo.853758

Tomita Tiberiu Florin. (1997). Some Remarks Concerning the Distribution of The Smarandache Function.  6 pages. http://doi.org/10.5281/zenodo.853774

M. Andrei, C. Dumitrescu, V. Seleacu, L. Tutescu, & St. Zamfir. (1997). Some Remarks on the Smarandache Function. 5 pages. http://doi.org/10.5281/zenodo.853834

Florian Luca. (1997). The Average Smarandache Function. 9 pages. http://doi.org/10.5281/zenodo.854294

Steven R. Finch. (2007). The Average Value of The Smarandache Function. 2 pages. http://doi.org/10.5281/zenodo.854298

Dhananjay P. Mehendale. (2007). The Classical Smarandache Function and a Formula for Twin Primes. 3 pages. http://doi.org/10.5281/zenodo.854306

Henry Ibstedt. (1993). The Florentin Smarandache Function S(n). 4 pages. http://doi.org/10.5281/zenodo.854308

Baohuai Shi. (2006). The hybrid mean value of the Smarandache function and the Mangoldt function. In Scientia Magna, Vol. 2 (2006), No. 4, pp. 98-100. http://doi.org/10.5281/zenodo.854310

Ion Balacenoiu. (1997). The Monotony of Smarandache Functions of First Kind. 7 pages. http://doi.org/10.5281/zenodo.854312

Kevin Ford. (2007). The Normal Behavior of the Smarandache Function. 6 pages. http://doi.org/10.5281/zenodo.854316

David Gorski. (2007). The Pseudo-Smarandache Function. 10 pages. http://doi.org/10.5281/zenodo.854322

Charles Ashbacher. (2007). The Pseudo-Smarandache Function and the Classical Functions of Number Theory.4 pages. http://doi.org/10.5281/zenodo.854326

Pal Gronas. (1992). The Solution of the Diophantine Equation. 3 pages. http://doi.org/10.5281/zenodo.854332

Charles Ashbacher. (2016). PalindromicNumbers of Pseudo-Smarandache Function. 5 pages. http://doi.org/10.5281/zenodo.854334

Wenpeng Zhang, & Ling Li. (2008). Two problems related to the Smarandache function. In Scientia Magna, Vol. 4 (2008), No. 2, pp.1-3. http://doi.org/10.5281/zenodo.854342

Marcela Popescu, Paul Popescu, Vasile Seleacu: On some numerical functions. Smarandache Function Journal 1995; 6 (1-2-3): 3-5

I. Balacenoiu, V. Seleacu, N. Virlan: Properties of the numerical function Fs. Smarandache Function Journal 1995; 6 (1-2-3): 6-10

Vasile Seleacu, Narcisa Virlan: On a limit of a sequence of a numerical function. Smarandache Function Journal 1995; 6 (1-2-3): 11-12

Emil Burton: On some series involving the Smarandache Function. Smarandache Function Journal 1995; 6 (1-2-3): 13-15

Ion Balacenoiu, Vasile Seleacu: Some properties of Smarandache Function of the type. Smarandache Function Journal 1995; 6 (1-2-3): 16-20

Charles Ashbacher: Some problems on Smarandache Function. Smarandache Function Journal 1995; 6 (1-2-3): 21-36

Ion Balacenoiu, Marcela Popescu, Vasile Seleacu: About the Smarandache Square's Complementary Function. Smarandache Function Journal 1995; 6 (1-2-3): 37-43

Tomita Tiberiu Florin: Some remarks concerning the distribution of the Smarandache Function. Smarandache Function Journal 1995; 6 (1-2-3): 44-49

E. Radescu, N. Radescu, C. Dumitrescu: Some elementary algebraic considerations inspired by the Smarandache Function. Smarandache Function Journal 1995; 6 (1-2-3): 50-54

Ion Balacenoiu, Constantin Dumitrescu: Smarandache Functions of the Second Kind. Smarandache Function Journal 1995; 6 (1-2-3): 55-58

Marcela Popescu, Paul Popescu: The problem of Lipschitz Condition. Smarandache Function Journal 1995; 6 (1-2-3): 59-63

Constantin Dumitrescu: A brief history of the "Smarandache Function" (III). Smarandache Function Journal 1995; 6 (1-2-3): 64-67

Thomas Martin: Proposed Problem. Smarandache Function Journal 1995; 6 (1-2-3): 68

J. Thompson: Proposed Problem. Smarandache Function Journal 1995; 6 (1-2-3): 69

Ken Tauscher: Proposed Problem of Number Theory. Smarandache Function Journal 1995; 6 (1-2-3): 70

L. Seagull: A generalization of a problem of Stuparu. Smarandache Function Journal 1995; 6 (1-2-3): 71

L. Seagull: An important formula to calculate the number of primes less than x. Smarandache Function Journal 1995; 6 (1-2-3): 72

M. Andrei, C. Dumitrescu, V. Seleacu, L. Tutescu, St. Zanfir: . Smarandache Function Journal 1993; 4-5: 1-5

Ion Balacenoiu: . Smarandache Function Journal 1993; 4-5: 6-13

Pal Gronas: Smarandache Function Journal 1993; 4-5: 14-16

E. Radescu, N. Radescu, C. Dumitrescu: . Smarandache Function Journal 1993; 4-5: 17-21

Pal Gronas: . Smarandache Function Journal 1993; 4-5: 21-23

J. R. Sutton: . Smarandache Function Journal 1993; 4-5: 24-26

J. R. Sutton: . Smarandache Function Journal 1993; 4-5: 27-31

Constantin Dumitrescu: . Smarandache Function Journal 1993; 4-5: 32-33

Henry Ibstedt: . Smarandache Function Journal 1993; 4-5: 34-35

J. Rodriguez: . Smarandache Function Journal 1993; 4-5: 36

Pal Gronas: . Smarandache Function Journal 1993; 4-5: 37

J. Rodriguez: . Smarandache Function Journal 1993; 4-5: 38

Pedro Melendez: . Smarandache Function Journal 1993; 4-5: 39

Pedro Melendez: . Smarandache Function Journal 1993; 4-5: 40

A. Stuparu, D. W. Sharpe: . Smarandache Function Journal 1993; 4-5: 41

T. Yau: . Smarandache Function Journal 1993; 4-5: 42

C. Dumitrescu: . Smarandache Function Journal 1993; 4-5: 43-44

T. Yau: . Smarandache Function Journal 1993; 4-5: 45

T. Yau: . Smarandache Function Journal 1993; 4-5: 46

T. Yau: . Smarandache Function Journal 1993; 4-5: 47

T. Yau: . Smarandache Function Journal 1993; 4-5: 48

Dr. Constantin Dumitrescu: . Smarandache Function Journal 1993; 2-3: 3-9

Mike Mudge: . Smarandache Function Journal 1993; 2-3: 10

Mike Mudge: . Smarandache Function Journal 1993; 2-3: 10

Jim Duncan: . Smarandache Function Journal 1993; 2-3: 11-12

Jim Duncan: . Smarandache Function Journal 1993; 2-3: 13-16

Jim Duncan: ... Smarandache Function Journal 1993; 2-3: 17-18

John C. McCarthy: . Smarandache Function Journal 1993; 2-3: 19-31

Mike Mudge: . Smarandache Function Journal 1993; 2-3: 32

Pal Gronas: . Smarandache Function Journal 1993; 2-3: 33

Pal Gronas: . Smarandache Function Journal 1993; 2-3: 34-35

John Sutton: . Smarandache Function Journal 1993; 2-3: 36-37

Henry Ibstedt: . Smarandache Function Journal 1993; 2-3: 38-71

R. Muller: . Smarandache Function Journal 1990; 1: 1-2

F. Smarandache: . Smarandache Function Journal 1990; 1: 3-17

F. Smarandache: . Smarandache Function Journal 1990; 1: 18-54

F. Smarandache: . Smarandache Function Journal 1990; 1: 55-61

F. Smarandache: . Smarandache Function Journal 1990; 1: 61-66

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