Smarandache Function

The smallest value $S(n)$ for a given $n$ for which $n\vert S(n)!$ ($n$ divides $S(n)$ Factorial). For example, the number 8 does not divide $1!$, $2!$, $3!$, but does divide $4!=4\cdot 3\cdot 2\cdot 1=8\cdot 3$, so $S(8)=4$. For a Prime $p$, $S(p)=p$, and for an Even Perfect Number $r$, $S(r)$ is Prime (Ashbacher 1997).

The Smarandache numbers for $n=1$, 2, ... are 1, 2, 3, 4, 5, 3, 7, 4, 6, 5, 11, ... (Sloane's A002034). Letting $a(n)$ denote the smallest value of $n$ for which $S(n)=1$, 2, ..., then $a(n)$ is given by 1, 2, 3, 4, 5, 9, 7, 32, 27, 25, 11, 243, ... (Sloane's A046021). Some values of $S(n)$ first occur only for very large $n$, for example, $S(59,049)=24$, $S(177,147)=27$, $S(134,217,728)=30$, $S(43,046,721)=36$, and $S(9,765,625)=45$. D. Wilson points out that if we let

$$I(n, p)={n-\Sigma(n, p)\over p-1},$$

be the power of the Prime $p$ in $n!$, where $\Sigma(n, p)$ is the sum of the base-$p$ digits of $n$, then it follows that

$$a(n) = \min p^{I(n-1, p)+1},$$

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

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

Tutescu (1996) conjectures that the Diophantine Equation $S(n)=S(n+1)$ 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.