The Smarandache function 
is the function that gives
the smallest positive integer 
such that 
, where 
is a positive integer
and 
is the factorial of 
. It was first considered by Lucas (1883), Neuberg (1887),
and Kempner (1918) and subsequently rediscovered by Smarandache (1980). 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 
is an integer satisfying 
(Kempner 1918). In addition,
 |
(6)
|
if 
is a Mersenne prime and 
is the corresponding even perfect
number (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
![nu=|_log_p[1+alpha(p-1)]_|,](/images/equations/SmarandacheFunction/NumberedEquation5.svg) |
(10)
|
where 
is the floor function. Now compute the sequences 
and 
according to the Euclidean
algorithm-like procedure
 |  |  |
(11)
|
 |  |  |
(12)
|
 |  |  | |
 |  |  |
(13)
|
 |  |  |
(14)
|
i.e., until the remainder 
. Set 
. At each step, 
is the integer part of 
and 
is the remainder. For example,
in the first step, 
and 
. Then
 |
(15)
|
(Kempner 1918).
The value of 
for general 
is then given by
![mu(p_1^(alpha_1)p_2^(alpha_2)...p_r^(alpha_r))=max[mu(p_1^(alpha_1)),mu(p_2^(alpha_2)),...,mu(p_r^(alpha_r))]](/images/equations/SmarandacheFunction/NumberedEquation7.svg) |
(16)
|
(Kempner 1918).
For all 
 |
(17)
|
where 
is the greatest prime factor of 
.
can be computed for 
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 asymptotic 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
 |
(18)
|
Ford (1999) and De Koninck and Doyon (2003) subsequently obtained asymptotic formulas, both with constants corrected by Ivić (2003). Ford's (1999) formula in its corrected form is
 |
(19)
|
where 
is the Dickman function and 
is the positive solution of
 |
(20)
|
Ivić (2003) subsequently showed that
 |
(21)
|
and, in terms of elementary functions,
![N(x)=xexp[-sqrt(2lnxlnlnx)(1+O(lnlnlnx/lnlnx))].](/images/equations/SmarandacheFunction/NumberedEquation13.svg) |
(22)
|
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
![a(k)=[gpf(k)]^(e+1),](/images/equations/SmarandacheFunction/NumberedEquation14.svg) |
(23)
|
for 
,
with 
,
where
![e=sum_(i=1)^(|_log_(gpf(k))(k-1)_|)|_(k-1)/([gpf(k)]^i)_|](/images/equations/SmarandacheFunction/NumberedEquation15.svg) |
(24)
|
(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
 |
(25)
|
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 (25) shows that the inverse Smarandache function 
always exists since for every 
there is an 
with 
(hence a smallest one 
), since 
for 
.
Sondow (2006) showed that 
unexpectedly arises in an irrationality
measure for e and conjectured that the inequality 
holds for almost
all 
,
where "for almost all" means except for a set of
asymptotic density zero. Li and Sha (2020) proved
the stronger result that 
holds for almost
all 
for each fixed real 
. The exceptions to the squared inequality
among 
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, the equivalent inequality 
also holds for almost
all 
.
Its exceptions among 
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
 |
(26)
|
is the exponent of the prime 
in 
for a nonnegative integer 
, where 
is the sum of the base-
digits of 
, then it follows for 
that
 |
(27)
|
where the minimum is taken over the primes 
dividing 
. This minimum appears to always be achieved when 
is the greatest prime
factor of 
.
." 5 Nov 2003. 
." Amer. Math. Monthly 101,
179, 1994.Kempner, A. J. "Miscellanea." Amer. Math.
Monthly 25, 201-210, 1918.Li, X. and Sha, M. "A Proof
of Sondow's Conjecture on the Smarandache Function." Amer. Math. Monthly 127,
939-943, 2020. 
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.