Smarandache function

In mathematics, the Smarandache function is a sequence or a number-theoretic function that is related to the faculty . Historically, it was first viewed by Édouard Lucas (1883), Joseph Neuberg (1887) and Aubrey J. Kempner (1918). In 1980 she was "rediscovered" by Florentin Smarandache .

Definition and values

The Smarandache function is defined as the smallest natural number for which the factorial of divides. ${\displaystyle \mu (n)}$ ${\displaystyle n}$ ${\displaystyle \mu (n)}$

Formal is the smallest natural number for which applies ${\displaystyle \mu (n)}$

${\displaystyle n\;|\;\mu (n)!}$

Examples

For example, if you are looking for the value , you have to look for the smallest of the numbers 1 !, 2 !, 3 !, ... which is divisible by 8. Since and and are not divisible by eight, but still, is . ${\displaystyle \mu (8)}$ ${\displaystyle \,1!=1}$ ${\displaystyle 2!=1\cdot 2=2}$ ${\displaystyle 3!=1\cdot 2\cdot 3=6}$ ${\displaystyle 4!=1\cdot 2\cdot 3\cdot 4=24=3\cdot 8}$ ${\displaystyle \,\mu (8)=4}$

However , since the number 7 is not one of the numbers 1 !, 2 !,…, 6! divides while she 7! trivially divides. ${\displaystyle \mu (7)=7}$

The first values ​​are:

n	1	2	3	4th	5	6th	7th	8th	9	10	11	12	13	14th	15th	16	17th	18th	19th	20th	21st	22nd	23	24	25th	26th	27	28	29
${\displaystyle \mu (n)}$	1 (*)	2	3	4th	5	3	7th	4th	6th	5	11	4th	13	7th	5	6th	17th	6th	19th	5	7th	11	23	4th	10	13	9	7th	29

(*) The value is also defined as 0 by some authors. ${\displaystyle \mu (1)}$

properties

Trivially, it applies

${\displaystyle \,\mu (n)\leq n,}$

because yes definitely shares. ${\displaystyle n}$ ${\displaystyle n!=n\cdot (n-1)!}$

A basic result is that equality in the above inequality occurs exactly for prime or : ${\displaystyle n}$ ${\displaystyle n=4}$

${\displaystyle \mu (n)=n\qquad \Leftrightarrow \qquad n{\text{ prim}}\quad {\text{oder}}\quad n=4}$

Proof:

${\displaystyle \Rightarrow }$: Be and not prime. Then is to show. Since is not prime, there are natural numbers with . If it were , it would and one would receive the contradiction . So is and therefore . If so followed , so and so , and you would have the contradiction again . Therefore it has to be and it follows . ${\displaystyle \mu (n)=n}$ ${\displaystyle n}$ ${\displaystyle n=4}$ ${\displaystyle n}$ ${\displaystyle 2\leq s\ \leq t<n}$ ${\displaystyle n=st}$ ${\displaystyle s<t}$ ${\displaystyle n=st|t!}$ ${\displaystyle \mu (n)\leq t<n}$ ${\displaystyle s=t}$ ${\displaystyle n=t^{2}}$ ${\displaystyle 2<t}$ ${\displaystyle t<2t<t^{2}=n}$ ${\displaystyle t<2t\leq n-1}$ ${\displaystyle n=t^{2}|t\cdot 2t|(2t)!|(n-1)!}$ ${\displaystyle \mu (n)<n}$ ${\displaystyle t=2}$ ${\displaystyle n=4}$

${\displaystyle \Leftarrow }$: If prime, then no number divides for , since per def. does not occur in. Therefore applies . is clear. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle m!}$ ${\displaystyle m<n}$ ${\displaystyle n}$ ${\displaystyle m!}$ ${\displaystyle \mu (n)=n}$ ${\displaystyle \mu (4)=4}$

Incidentally, this results in , the number of prime numbers less than or equal to and the integer function : ${\displaystyle \pi (x)}$ ${\displaystyle x}$

${\displaystyle \pi (x)=-1+\sum _{k=2}^{x}\left\lfloor {\frac {\mu (k)}{k}}\right\rfloor }$.

According to Paul Erdős, agrees with the greatest prime factor of for asymptotically almost all , i.e. H. the number of numbers less than or equal to which this does not apply is o (n) . ${\displaystyle \mu (n)}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$

Generally also applies

${\displaystyle \,\mu (n!)=n}$

and

${\displaystyle \mu (n)\geq \mathrm {gpf} (n)}$

where stands for the largest prime factor of . ${\displaystyle {\rm {gpf}}}$ ${\displaystyle n}$

The following applies in general

${\displaystyle \mu \left(p_{1}^{\alpha _{1}}\cdot p_{2}^{\alpha _{2}}\cdot \ldots \cdot p_{n}^{\alpha _{n}}\right)=\max \left[\mu \left(p_{1}^{\alpha _{1}}\right),\mu \left(p_{2}^{\alpha _{2}}\right),\ldots ,\mu \left(p_{n}^{\alpha _{n}}\right)\right]}$

For (even) perfect numbers we also have ( ) ${\displaystyle n}$ ${\displaystyle k\in \mathbb {N} ,p{\text{ prim}}}$

${\displaystyle \mu (n)=\mu (2^{k-1}\cdot (2^{k}-1))=2^{k}-1=p}$

Modifications

Pseudosmarandache function

The pseudosmarandache function is the smallest integer for which ${\displaystyle Z(n)}$

${\displaystyle n\;\;{\text{teilt}}\;\;1+2+3+\cdots +Z(n),}$

so the smallest natural for which applies ${\displaystyle n}$

${\displaystyle n\;\;\left|\;\;{\frac {Z(n)(Z(n)+1)}{2}}\right.}$

(see also triangular number , Gaussian sum formula )

The first values ​​are

1, 3, 2, 7, 4, 3, 6, 15, 8, 4, 10, 8, 12, 7, 5, 31, 16, 8, 18, 15, ... (sequence A011772 in OEIS )

Some features:

• ${\displaystyle {\sqrt {n}}<Z(n)\leq 2n-1}$
• ${\displaystyle Z(n)\leq n-1\qquad {\text{für ungerade }}n}$
• ${\displaystyle Z(2^{k})=2^{k+1}-1\,}$
• ${\displaystyle {\frac {Z(n+1)}{Z(n)}}{\text{ und }}{\frac {Z(n-1)}{Z(n)}}{\text{ und }}{\frac {Z(2n)}{Z(n)}}}$ are unlimited at the top
• ${\displaystyle {\frac {n}{Z(n)}}=k\;(k\in \mathbb {Z} ,k\geq 2)}$ has infinite solutions for ${\displaystyle n}$
• ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{Z(n)^{\alpha }}}}$ converges for all ${\displaystyle \alpha >1}$

Smarandache dual faculty function

If you replace the faculty with the double faculty in the definition

${\displaystyle n!!={\begin{cases}n\cdot (n-2)\cdot (n-4)\cdot \ldots \cdot 2&{\text{für }}n{\text{ gerade,}}\\n\cdot (n-2)\cdot (n-4)\cdot \ldots \cdot 1&{\text{für }}n{\text{ ungerade,}}\end{cases}}}$

so is ${\displaystyle \mathrm {Sdf} (n)}$

the smallest natural number that is divisible by . ${\displaystyle \mathrm {Sdf} (n)!!}$

The first values ​​for are ${\displaystyle \mathrm {Sdf} (n)}$

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 6, 13, 14, 5, 6, ... (sequence A007922 in OEIS )

Smarandache function with primorial

The primorial (also prime faculty ) is the product of the prime numbers less than or equal to the given number. The Smarandache Near-to-Primorial Function of is then the smallest prime number for which , or is divisible by . ${\displaystyle p_{\#}}$ ${\displaystyle n}$ ${\displaystyle p_{\#}+1}$ ${\displaystyle p_{\#}-1}$ ${\displaystyle p_{\#}}$ ${\displaystyle n}$

Smarandache Kurepa function and Smarandache Wagstaff function

For the Smarandache-Kurepa function , the faculty is not converted to a double faculty, but to the following function: ${\displaystyle \mathrm {SK} (n)}$

${\displaystyle f(n)=\sum _{k=0}^{n-1}k!=0!+1!+2!+\ldots +(n-1)!}$

For prime is analogously the smallest natural number, so that is divisible by . ${\displaystyle p}$ ${\displaystyle \mathrm {SK} (p)}$ ${\displaystyle f(\mathrm {SK} (p))}$ ${\displaystyle p}$

The first values ​​are 2, 4, 6, 6, 5, 7, 7, 12, 22, 16, 55 and form the sequence A049041 in OEIS .

The Smarandache-Wagstaff function is used instead

${\displaystyle f(n)=\sum _{k=1}^{n}k!=1!+2!+\ldots +n!}$

Smarandache ceil function

The Smarandache-Wagstaff function k-th order finally is defined as the smallest natural number for which by is divisible. ${\displaystyle S_{k}(n)}$ ${\displaystyle \,[S_{k}(n)]^{k}}$ ${\displaystyle n}$

The first values:

${\displaystyle k}$	${\displaystyle S_{k}(n)}$
1	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...	${\displaystyle n}$
2	1, 2, 3, 2, 5, 6, 7, 4, 3, 10, 11, 6, 13, 14, 15, ...	(Follow A019554 in OEIS )
3	1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, ...	(Follow A019555 in OEIS )
4th	1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, ...	(Follow A053166 in OEIS )

additional

• Tutescu assumed that for two consecutive numbers the values ​​of the Smarandache function are always different:

${\displaystyle \mu (n)\not =\mu (n+1)\qquad \quad {\text{für alle }}n}$

The assumption was recalculated until it was confirmed. ${\displaystyle 10^{9}}$

• There is quite a wide variety of convergent series that use the Smarandache function. Such limit values ​​are often referred to as Smarandache constants - not to be confused with the Smarandache constant in the generalized Andrican conjecture .

The series of reciprocal values ​​of the faculties of the Smarandache function converges ( first Smarandache constant ):

${\displaystyle \sum _{n=2}^{\infty }{\frac {1}{\mu (n)!}}=1{,}09317\dots }$(Follow A048799 in OEIS )

literature

• Kenichiro Kashihara: Comments and topics on Smarandache notions and problems . (PDF) Erhus University Press 1996, ISBN 1-87958-555-3 .
• Norbert Hungerbühler, Ernst Specker: A Generalization of the Smarandache Function to Several Variables . (PDF) In: Electronic Journal of Combinatorical Number Theory , 6, 2006, # A23.
• C. Dumitrescu, N. Virlan, St. Zamfir, E. Radescu, N. Radescu, F.Smarandache: Smarandache Type Function Obtained by Duality. In: Studii si Cercetari Stiintifice , Seria: Matematica, University of Bacau, No. 9, 1999, pp. 49-72, arxiv : 0706.2858 .
• Sebastian Martin Ruiz, ML Perez: Properties and Problems related to Smarandache Type Functions. In: Mathematics Magazine for grades 1-12 , 2/2004, pp. 46-53, arxiv : math / 0407479 .

Web links

• Eric W. Weisstein : Smarandache Function . In: MathWorld (English).
• The Smarandache Function Journal , fs.gallup.unm.edu - Vol. 1 (PDF; 1.7 MB), Vol. 6 (PDF; 2.6 MB)
• and Smarandache Notions Journal - Vol. 7 (PDF; 5.4 MB), Vol. 8 (PDF; 8.8 MB), Vol. 9 (PDF; 5.3 MB), Vol. 10 (PDF; 7.3 MB), Vol. 11 (PDF; 10.8 MB), Vol. 12 (PDF; 12.5 MB), Vol. 13 (PDF; 11.1 MB)

Individual evidence

1. ↑ E. Lucas : Question No. 288 . In: Mathesis , 3, 1883, p. 232
2. ↑ J. Neuberg : Solutions de questions proposées, Question No. 288 . In: Mathesis , 7, 1887, pp. 68-69
3. ^ Aubrey J. Kempner: Miscellanea . In: American Mathematical Monthly , 25, 1918, pp. 201-210, doi: 10.2307 / 2972639
4. ↑ Florentin Smarandache : A Function in Number Theory. In: To. Univ. Timişoara , Ser. St. Mat., 18, 1980, pp. 79-88. arxiv : math / 0405143
5. ↑ Follow A002034 in OEIS
6. ↑ Sebastián Martín Ruiz: Smarandache's function applied to perfect numbers . In: Smarandache Notions Journal , Vol. 10, Spring 1999, p. 114. arxiv : math / 0406241
7. ↑ RGE Pinch: arxiv : math / 0504118 in arXiv , April 6, 2005
8. ↑ Eric W. Weisstein : Smarandache Near-to-Primorial Function . In: MathWorld (English).
9. ↑ Eric W. Weisstein : Smarandache-Kurepa Function . In: MathWorld (English).
10. ↑ Eric W. Weisstein : Smarandache-Wagstaff Function . In: MathWorld (English).
11. ↑ Eric W. Weisstein : Smarandache Ceil Function . In: MathWorld (English).
12. ↑ L. Tutescu: On a Conjecture Concerning the Smarandache Function. Abstracts of Papers Presented to the American Mathematical Society 17, p. 583, 1996