## Smarandache Function

The smallest value for a given for which ( divides Factorial). For example, the number 8 does not divide , , , but does divide , so . For a Prime , , and for an Even Perfect Number , is Prime (Ashbacher 1997).

The Smarandache numbers for , 2, ... are 1, 2, 3, 4, 5, 3, 7, 4, 6, 5, 11, ... (Sloane's A002034). Letting denote the smallest value of for which , 2, ..., then is given by 1, 2, 3, 4, 5, 9, 7, 32, 27, 25, 11, 243, ... (Sloane's A046021). Some values of first occur only for very large , for example, , , , , and . D. Wilson points out that if we let

be the power of the Prime in , where is the sum of the base- digits of , then it follows that

where the minimum is taken over the Primes dividing . This minimum appears to always be achieved when is the Greatest Prime Factor of .

The incrementally largest values of are 1, 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, ... (Sloane's A046022), which occur for , 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, ... (Sloane's A046023), i.e., the values where .

Tutescu (1996) conjectures that the Diophantine Equation has no solution.

See also Factorial, Greatest Prime Factor, Pseudosmarandache Function, Smarandache Ceil Function, Smarandache Constants, Smarandache-Kurepa Function, Smarandache Near-to-Primorial Function, Smarandache-Wagstaff Function

References

Ashbacher, C. An Introduction to the Smarandache Function. Cedar Rapids, IA: Decisionmark, 1995.

Ashbacher, C. Problem 4616.'' School Sci. Math. 97, 221, 1997.

Begay, A. Smarandache Ceil Functions.'' Bulletin Pure Appl. Sci. India 16E, 227-229, 1997.

Dumitrescu, C. and Seleacu, V. The Smarandache Function. Vail, AZ: Erhus University Press, 1996.

Functions in Number Theory.'' http://www.gallup.unm.edu/~smarandache/FUNCT1.TXT.

Ibstedt, H. Surfing on the Ocean of Numbers--A Few Smarandache Notions and Similar Topics. Lupton, AZ: Erhus University Press, pp. 27-30, 1997.

Sandor, J. On Certain Inequalities Involving the Smarandache Function.'' Abstracts of Papers Presented to the Amer. Math. Soc. 17, 583, 1996.

Sloane, N. J. A. Sequences A046021, A046022, A046023, and A002034/M0453 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Smarandache, F. Collected Papers, Vol. 1. Bucharest, Romania: Tempus, 1996.

Smarandache, F. Collected Papers, Vol. 2. Kishinev, Moldova: Kishinev University Press, 1997.

Tutescu, L. On a Conjecture Concerning the Smarandache Function.'' Abstracts of Papers Presented to the Amer. Math. Soc. 17, 583, 1996.