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 .

REFERENCES:

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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.

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

Finch, S. "Moments of the Smarandache Function." Smarandache Notions J. 11, 140-141, 2000. https://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. https://www.gallup.unm.edu/~smarandache/SNBook10.pdf.

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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. https://arxiv.org/abs/math.NT/0311056.

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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.