I Need A Bigger Computer Or a Physist To This Sequence Of Building A Barrel. First Proof!! Hello Amy!! You Are In England, Run This Over To Department of Physics, University of Oxford, It Will Be Like The Old Days, You Know Like The Movies With Those Actors Acting The Parts Of The Real People!! Don't Forget My Sweatshirt :)

 sequence 

10 12 15 17 

10 12 15 17.2 

2 3 10 12 15 17.2 

2 3 10 12 15 17.2 

10 12 15 17 

10 12 15 17.2 

2 3 10 12 15 17.2 

2 3 10 12 15 17.2 

10 12 15 17 

10 12 15 17.2 

2 3 10 12 15 17.2 

2 3 10 12 15 17.2 

10 12 15 17 

10 12 15 17.2 

10 12 17.2

10 12 17

2 3 10 12

17.2 10 12

12 10 17.2

2 3 17.3 

3 2 17.12

12 2 17.1

3 2 12.5

3 2 12.5 2 

13 5 = done.

Smarandache Function

The Smarandache function is the function first considered by Lucas (1883), Neuberg (1887), and Kempner (1918) and subsequently rediscovered by Smarandache (1980) that gives the smallest value for a given at which (i.e., divides factorial). For example, the number 8 does not divide , , , but does divide , so .
For , 2, ..., is given by 1, 2, 3, 4, 5, 3, 7, 4, 6, 5, 11, ... (OEIS A002034), where it should be noted that Sloane defines , while Ashbacher (1995) and Russo (2000, p. 4) take . The incrementally largest values of are 1, 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, ... (OEIS A046022), which occur at the values where . The incrementally smallest values of relative to are = 1, 1/2, 1/3, 1/4, 1/6, 1/8, 1/12, 3/40, 1/15, 1/16, 1/24, 1/30, ... (OEIS A094404 and A094372), which occur at , 6, 12, 20, 24, 40, 60, 80, 90, 112, 120, 180, ... (OEIS A094371).
Formulas exist for immediately computing for special forms of . The simplest cases are

			(1)
			(2)
			(3)
			(4)
			(5)

where is a prime, are distinct primes, , and (Kempner 1918). In addition,

(6)

if is the th even perfect number and is the corresponding Mersenne prime (Ashbacher 1997; Ruiz 1999a). Finally, if is a prime number and an integer, then

(7)

(Ruiz 1999b).
The case for is more complicated, but can be computed by an algorithm due to Kempner (1918). To begin, define recursively by

(8)

with . This can be solved in closed form as

(9)

Now find the value of such that , which is given by

(10)

where is the floor function. Now compute the sequences and according to the Euclidean algorithm-like procedure

			(11)
			(12)
			(13)
			(14)
			(15)

i.e., until the remainder . At each step, is the integer part of and is the remainder. For example, in the first step, and . Then

(16)

(Kempner 1918).
The value of for general is then given by

(17)

(Kempner 1918).
For all

(18)

where is the greatest prime factor of .
can be computed by finding and testing if divides . If it does, then . If it doesn't, then and Kempner's algorithm must be used. The set of for which (i.e., does not divide ) has density zero as proposed by Erdős (1991) and proved by Kastanas (1994), but for small , there are quite a large number of values for which . The first few of these are 4, 8, 9, 12, 16, 18, 24, 25, 27, 32, 36, 45, 48, 49, 50, ... (OEIS A057109). Letting denote the number of positive integers such that , Akbik (1999) subsequently showed that

(19)

This was subsequently improved by Ford (1999) and De Koninck and Doyon (2003), the former of which is unfortunately incorrect. Ford (1999) proposed the asymptotic formula

(20)

where is the Dickman function, is defined implicitly through

(21)

and the constant needs correction (Ivić 2003). Ivić (2003) subsequently showed that

(22)

and, in terms of elementary functions,

(23)

Tutescu (1996) conjectured that never takes the same value for two consecutive arguments, i.e., for any . This holds up to at least (Weisstein, Mar. 3, 2004).
Multiple values of can have the same value of , as summarized in the following table for small .

	such that
1	1
2	2
3	3, 6
4	4, 8, 12, 24
5	5, 10, 15, 20, 30, 40, 60, 120
6	9, 16, 18, 36, 45, 48, 72, 80, 90, 144, 180, 240, 360, 720

Let denote the smallest inverse of , i.e., the smallest for which . Then is given by

(24)

where

(25)

(J. Sondow, pers. comm., Jan. 17, 2005), where is the greatest prime factor of and is the floor function. For , 2, ..., is given by 1, 2, 3, 4, 5, 9, 7, 32, 27, 25, 11, 243, ... (OEIS A046021). Some values of first occur only for very large . The sequence of incrementally largest values of is 1, 2, 3, 4, 5, 9, 32, 243, 4096, 59049, 177147, 134217728, ... (OEIS A092233), corresponding to , 2, 3, 4, 5, 6, 8, 12, 16, 24, 27, 32, ... (OEIS A092232).
To find the number of for which , note that by definition, is a divisor of but not of . Therefore, to find all for which has a given value, say all with , take the set of all divisors of and omit the divisors of . In particular, the number of for which for is exactly

(26)

where denotes the number of divisors of , i.e., the divisor function . Therefore, the numbers of integers with , 2, ... are given by 1, 1, 2, 4, 8, 14, 30, 36, 64, 110, ... (OEIS A038024).
In particular, equation (26) shows that the inverse Smarandache function always exists since for every there is an with (hence a smallest one a(n)), since for .
Sondow (2006) showed that unexpectedly arises in an irrationality bound for e, and conjectures that the inequality holds for almost all , where "for almost all" means except for a set of density zero. The exceptions are 2, 3, 6, 8, 12, 15, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, ... (OEIS A122378).
Since for almost all (Erdős 1991, Kastanas 1994), where is the greatest prime factor, an equivalent conjecture is that the inequality holds for almost all . The exceptions are 2, 3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, ... (OEIS A122380).
D. Wilson points out that if

(27)

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

(28)

where the minimum is taken over the primes dividing . This minimum appears to always be achieved when is the greatest prime factor of .

SEE ALSO: Factorial, Greatest Prime Factor, Pseudosmarandache Function, Smarandache Ceil Function, Smarandache Constants, Smarandache-Kurepa Function, Smarandache Near-to-Primorial Function, Smarandache-Wagstaff Function Portions of this entry contributed by Jonathan Sondow (author's link)

REFERENCES:

Akbik, S. "On a Density Problem of Erdős." Int. J. Math. Sci. 22, 655-658, 1999.

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.

De Koninck, J.-M. and Doyon, N. "On a Thin Set of Integers Involving the Largest Prime Factor Function." Int. J. Math. Math. Sci., No. 19, 1185-1192, 2003.

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

Erdős, P. "Problem 6674." Amer. Math. Monthly 98, 965, 1991.

Finch, S. "The Average Value of the Smarandache Function." Smarandache Notions J. 10, 95-96, 1999. http://www.gallup.unm.edu/~smarandache/SNBook10.pdf.

Finch, S. "Moments of the Smarandache Function." Smarandache Notions J. 11, 140-141, 2000. http://www.gallup.unm.edu/~smarandache/SNBook11.pdf.

Finch, S. R. "Golomb-Dickman Constant." §5.4 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 284-292, 2003.

Ford, K. "The Normal Behavior of the Smarandache Function." Smarandache Notions J. 10, 81-86, 1999. http://www.gallup.unm.edu/~smarandache/SNBook10.pdf.

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

Hungerbühler, N. and Specker, E. "A Generalization of the Smarandache Function to Several Variables." Integers: Electronic J. Combin. Number Th. 6, #A23, 2006 http://www.integers-ejcnt.org/vol6.html.

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

Ivić, A. "On a Problem of Erdős Involving the Largest Prime Factor of ." 5 Nov 2003. http://arxiv.org/abs/math.NT/0311056.

Kastanas, I. "Solution to Problem 6674: The Smallest Factorial That Is a Multiple of ." Amer. Math. Monthly 101, 179, 1994.

Kempner, A. J. "Miscellanea." Amer. Math. Monthly 25, 201-210, 1918.

Lucas, E. "Question Nr. 288." Mathesis 3, 232, 1883.

Neuberg, J. "Solutions de questions proposées, Question Nr. 288." Mathesis 7, 68-69, 1887.

Ruiz, S. M. "Smarandache Function Applied to Perfect Numbers." Smarandache Notions J. 10, 114-155, 1999a.

Ruiz, S. M. "A Result Obtained Using Smarandache Function." Smarandache Notions J. 10, 123-124, 1999b.

Russo, F. A Set of New Smarandache Functions, Sequences, and Conjectures in Numer Theory. Lupton, AZ: American Research Press, 2000.

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 A002034/M0453, A038024, A046021, A046022, A057109, A092232, A092233, A094371, A094372, A094404, A122378, and A122380 in "The On-Line Encyclopedia of Integer Sequences."

Smarandache, F. "A Function in Number Theory." Analele Univ. Timisoara, Ser. St. Math. 43, 79-88, 1980.

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

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

Sondow, J. "A Geometric Proof that Is Irrational and a New Measure of Its Irrationality." Amer. Math. Monthly 113, 637-641, 2006.

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

Referenced on Wolfram|Alpha: Smarandache Function

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Sondow, Jonathan and Weisstein, Eric W. "Smarandache Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SmarandacheFunction.html

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