Smarandache function

Smarandache function

Smarandache function

The

Smarandache function

S

:

ℤ

+

→

ℤ

+

is defined as follows:

S

⁢

(

n

)

is the smallest

integer

such that

S

⁢

(

n

)

!

is

divisible

by

n

. For example, the number 8 does not divide

1

!

,

2

!

,

3

!

, but does divide

4

!

. Therefore

S

⁢

(

8

)

=

4

. Another study of

S

⁢

(

n

)

has been published by

http://genealogy.math.ndsu.nodak.edu/html/id.phtml?id=12376

Aubrey J. Kempner in 1918, and later the function

S

⁢

(

n

)

has been rediscovered and studied by

Florentin Smarandache

in 1980. A profound study of this function would contribute to the study of

prime numbers

in accordance with the following

property

: if

p

is a number greater than

4

, then

p

is a prime if and only if

S

⁢

(

p

)

=

p

. The values of

S

⁢

(

n

)

for

n

=

1

,

2

,

3

,

…

are given by Sloane’s (

http://

planetmath

.org/NeilSloane

) OEIS

http://www.research.att.com/ njas/

sequences

/?q=A002034

A002034.

A list of sixteen s denoted

s

1

to

s

16

have been defined with the use of the Smarandache function

S

⁢

(

n

)

, and they should not be confused with

the

Smarandache constant

, which is the smallest solution to the

generalized Andrica conjecture

.

The first Smarandache constant (Sloane’s (

http://planetmath.org/NeilSloane

) OEIS

http://www.research.att.com/ njas/sequences/?q=A048799

A048799) is defined as

s

1

=

∑

n

=

2

∞

(

S

⁢

(

n

)

!

)

-

1

≈

1.09317

⁢

…

The second Smarandache constant

s

2

(Sloane’s (

http://planetmath.org/NeilSloane

) OEIS

http://www.research.att.com/ njas/sequences/?q=A048834

A048834) is defined as

s

2

=

∑

n

=

2

∞

S

⁢

(

n

)

n

!

≈

1.71400629359162

⁢

…

and it is an

irrational number

.

The third Smarandache constant

s

3

(Sloane’s (

http://planetmath.org/NeilSloane

) OEIS

http://www.research.att.com/ njas/sequences/?q=A048835

A048835) is defined as

s

3

=

∑

n

=

2

∞

(

∏

i

=

2

n

S

⁢

(

i

)

)

-

1

≈

0.719960700043

⁢

…

.

The series

s

4

⁢

(

α

)

=

∑

n

=

2

∞

n

α

⁢

(

∏

i

=

2

n

S

⁢

(

i

)

)

-

1

converges

for a fixed

real number

α

≥

1

. Since

s

4

is a function of

α

it is not a single constant, but an infinite list of them. The values for small

α

have been computed:

s

4

⁢

(

1

)

≈

1.72875760530223

⁢

…

(Sloane’s (

http://planetmath.org/NeilSloane

) OEIS

http://www.research.att.com/ njas/sequences/?q=A048836

A048836).

s

4

⁢

(

2

)

≈

4.50251200619296

⁢

…

(Sloane’s (

http://planetmath.org/NeilSloane

) OEIS

http://www.research.att.com/ njas/sequences/?q=A048837

A048837).

s

4

⁢

(

3

)

≈

13.0111441949445

⁢

…

(Sloane’s (

http://planetmath.org/NeilSloane

) OEIS

http://www.research.att.com/ njas/sequences/?q=A048838

A048838).

The fifth Smarandache constant

s

5

=

∑

n

=

1

∞

(

-

1

)

n

-

1

⁢

S

⁢

(

n

)

n

!

converges to an irrational number.

Burton in 1995 showed that the series

s

6

=

∑

n

=

2

∞

S

⁢

(

n

)

(

n

+

1

)

!

converges and is

bounded

by

0.218282

<

s

6

<

0.5

.

Dumitrescu and Seleacu in 1996 showed that the series

s

7

⁢

(

r

)

=

∑

n

=

r

∞

S

⁢

(

n

)

(

n

+

r

)

!

and

s

8

⁢

(

r

)

=

∑

n

=

r

∞

S

⁢

(

n

)

(

n

-

r

)

!

converge for

r

∈

ℤ

+

.

The same authors show that the series

s

9

=

∑

n

=

2

∞

(

∑

i

=

2

n

S

⁢

(

i

)

i

!

)

-

1

is

convergent

.

The series

s

10

⁢

(

α

)

=

∑

n

=

2

∞

(

S

⁢

(

n

)

)

-

α

⁢

(

S

⁢

(

n

)

!

)

-

1

2

and

s

11

⁢

(

α

)

=

∑

n

=

2

∞

(

S

⁢

(

n

)

)

-

α

⁢

[

(

S

⁢

(

n

)

-

1

)

!

]

-

1

2

converge for

α

>

1

. These two series also define an infinite list of constants.

If

f

:

ℕ

→

ℝ

is a function satisfying the condition

f

⁢

(

t

)

≤

c

t

α

⁢

d

⁢

(

t

!

)

-

d

⁢

(

(

t

-

1

)

!

)

, where

t

is a

positive integer

,

d

denotes the

divisor function

, and the given constants

α

>

1

,

c

>

1

, then the series

s

12

⁢

(

f

)

=

∑

n

=

1

∞

f

⁢

(

S

⁢

(

n

)

)

is convergent.

The series

s

13

=

∑

n

=

1

∞

(

∏

k

=

1

n

S

⁢

(

k

)

!

)

-

1

n

is convergent.

The series

s

14

⁢

(

α

)

=

∑

n

=

1

∞

(

S

⁢

(

n

)

!

)

-

3

2

⁢

(

log

⁡

S

⁢

(

n

)

)

-

α

is convergent for

α

>

1

.

The series

s

15

=

∑

n

=

1

∞

2

n

S

⁢

(

2

n

)

!

is convergent.

The series

s

16

⁢

(

α

)

=

∑

n

=

1

∞

S

⁢

(

n

)

n

1

+

α

is convergent for

α

>

1

.

References

1. Dumitrescu C, Popescu M, Seleacu V, Tilton H (1996). The Smarandache Function in

Number Theory

. Erhus University Press. ISBN 1879585472.

2. Ashbacher C, Popescu M (1995). An Introduction to the Smarandache Function. Erhus University Press. ISBN 1879585499.

3. Tabirca S, Tabirca T, Reynolds K, Yang LT (2004).

http://dx.doi.org/10.1109/ISPDC.2004.15

”Calculating Smarandache function in parallel”. Parallel and

Distributed Computing

, 2004. Third International Symposium on

Algorithms

, Models and Tools for

Parallel Computing

on Heterogeneous Networks,: pp.79-82.

4. Kempner AJ (1918). ”Miscellanea”.

http://www.jstor.org/view/00029890/di991004/99p1446d/0

The American Mathematical Monthly 25: 201-210.

5. Mehendale DP (2005).

http://arxiv.org/abs/math/0502384

The Classical Smarandache Function and a Formula for

Twin Primes

.

6. Smarandache F (1980). ”A Function in Number Theory”. Analele Univ. Timisoara, Ser. St. Math. 43: 79-88.

7. Smarandache F.

http://www.gallup.unm.edu/ smarandache/CONSTANT.TXT

Constants Involving the Smarandache Function.

8. Muller R (1990). ”Editorial”.

http://www.gallup.unm.edu/ smarandache/SFJ1.pdf

Smarandache Function Journal

1: 1.

9. Cojocaru I, Cojocaru S (1996). ”The First Constant of Smarandache”.

http://www.gallup.unm.edu/ smarandache/SNJ7.pdf

Smarandache Notions Journal 7: 116-118.

10. Cojocaru I, Cojocaru S (1996). ”The Second Constant of Smarandache”.

http://www.gallup.unm.edu/ smarandache/SNJ7.pdf

Smarandache Notions Journal 7: 119-120.

11. Cojocaru I, Cojocaru S (1996). ”The Third and Fourth Constants of Smarandache”.

http://www.gallup.unm.edu/ smarandache/SNJ7.pdf

Smarandache Notions Journal 7: 121-126.

12. Sandor J (1997). ”On The Irrationality Of Certain Alternative Smarandache Series”.

http://www.gallup.unm.edu/ smarandache/SNJ8.pdf

Smarandache Notions Journal 8: 143-144.

13. Burton E (1995). ”On Some Series Involving the Smarandache Function”.

http://www.gallup.unm.edu/ smarandache/SFJ6.pdf

Smarandache Function Journal 6: 13-15.

14. Burton E (1996). ”On Some

Convergent Series

”.

http://www.gallup.unm.edu/ smarandache/SNJ7.pdf

Smarandache Notions Journal 7 (1-3): 7-9.

Title

Smarandache function

Canonical

name

SmarandacheFunction

Date of creation

2013-03-22 17:04:15

Last modified on

2013-03-22 17:04:15

Owner

dankomed (17058)

Last modified by

dankomed (17058)

Numerical id

47

Author

dankomed (17058)

Entry type

Definition

Classification

msc 11A41

Related topic

GeneralizedAndricaConjecture

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