Smarandache Function
The smallest value 
for a given 
for which 
( divides Factorial). For example, the number8 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 followsthat
where the minimum is taken over the Primes
dividing . This minimum appears to always be achieved when is theGreatest 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 occurfor , 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.
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, andA002034/M0453in ``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.
- Permanent
- Permil
- Permutation
- Permutation Group
- Permutation Matrix
- Permutation Pseudotensor
- Permutation Symbol
- Permutation Tensor
- Peron Integral
- Perpendicular
- Perpendicular Bisector
- Perpendicular Foot
- Perrin Pseudoprime
- Perrin Sequence
- Perron-Frobenius Operator
- Perron-Frobenius Theorem
- Perron's Theorem
- Persistence
- Persistent Number
- Persistent Process
- Perspective
- Perspective Axis
- Perspective Center
- Perspective Collineation
- Perspective Triangles