Smarandache constants

In number theory, one speaks of Smarandache constants (according to Florentin Smarandache ) in two contexts, one with the Andrican conjecture and the other with the Smarandache function . The two definitions have nothing in common apart from their namesake.

1. ext

If the -th prime number denotes, Andrica's conjecture says that for all ${\displaystyle p_{n}}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle {\sqrt {p_{n+1}}}-{\sqrt {p_{n}}}<1.}$ This assumption can be generalized as follows: ${\displaystyle p_{n+1}^{a}-p_{n}^{a}<1\qquad \quad \mathrm {f{\ddot {u}}r\;alle} \;a<a_{0}.}$ This upper limit for , approximately , is often referred to as the Smarandache constant. is solution of the equation . ${\displaystyle a_{0}}$ ${\displaystyle 0{,}56714813...}$ ${\displaystyle a_{0}}$ ${\displaystyle p_{31}^{x}-p_{30}^{x}=127^{x}-113^{x}=1}$

2.

The Smarandache function is defined as follows: ${\displaystyle \mu (n)}$ ${\displaystyle \mu (n)}$is the smallest natural number that is divisible by . ${\displaystyle \mu (n)!}$ ${\displaystyle n}$ 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; that is 4! = 24 = 3 * 8, therefore is . Various convergent series that use the values ​​of this function have now been examined. Such limit values ​​are then called first, second, ... Smarandache constants. ${\displaystyle \mu (8)}$ ${\displaystyle \mu (8)=4}$

Smarandache constants

The first Smarandache constant is defined by

${\displaystyle s_{1}=\sum _{n=2}^{\infty }{\frac {1}{\mu (n)!}}=1{,}093170459...}$

Their convergence with and the Euler number of readily appreciated as the upper limit: . ${\displaystyle \mu (n)\leq n}$ ${\displaystyle \sum {\frac {1}{\mu (n)!}}<\sum {\frac {1}{n!}}=e}$

The decimal places form the sequence A048799 in OEIS .

The second Smarandache constant is

${\displaystyle s_{2}=\sum _{n=2}^{\infty }{\frac {\mu (n)}{n!}}=1{,}7140062935916...}$

For this one must also prove that it is irrational ; it is sequence A048834 in OEIS .

The third Smarandache constant is then

${\displaystyle s_{3}=\sum _{n=2}^{\infty }{\frac {1}{\mu (2)\cdot \mu (3)\cdots \mu (n)}}=0{,}7199607000437...}$

Your decimal places result in the sequence A048835 in OEIS .

Furthermore, the following series converges for all real numbers : ${\displaystyle \alpha \geq 1}$

${\displaystyle s_{4}(\alpha )=\sum _{n=2}^{\infty }{\frac {n^{\alpha }}{\mu (2)\cdot \mu (3)\cdots \mu (n)}}}$

The first values ​​for natural : ${\displaystyle \alpha }$

${\displaystyle \alpha }$	${\displaystyle S_{4}(\alpha )}$
1	1.7287576053 ...	(Follow A048836 in OEIS )
2	4.5025120061 ...	(Follow A048837 in OEIS )
3	13.011144194 ...	(Follow A048838 in OEIS )

Other authors proved that

${\displaystyle s_{5}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}\mu (n)}{n!}}}$

also has a limit. The next constant

${\displaystyle s_{6}=\sum _{n=2}^{\infty }{\frac {\mu (n)}{(n+1)!}},}$

converges to a value . ${\displaystyle 0{,}218282<s_{6}<0{,}5}$

More generally, they even converge

${\displaystyle s_{7}=\sum _{n=2}^{\infty }{\frac {\mu (n)}{(n+k)!}}\quad {\text{und}}\quad s_{8}=\sum _{n=2}^{\infty }{\frac {\mu (n)}{(n-k)!}}}$

for natural (or whole ) . ${\displaystyle k\not =0}$

Also converges

${\displaystyle s_{9}=\sum _{n=2}^{\infty }{\frac {1}{{\frac {\mu (2)}{2!}}+{\frac {\mu (3)}{3!}}+\cdots +{\frac {\mu (n)}{n!}}}}.}$

Two more rows are

${\displaystyle s_{10}(\alpha )=\sum _{n=2}^{\infty }{\frac {1}{\mu (n)^{\alpha }{\sqrt {\mu (n)!}}}}}$

and

${\displaystyle s_{11}(\alpha )=\sum _{n=2}^{\infty }{\frac {1}{\mu (n)^{\alpha }{\sqrt {(\mu (n)+1)!}}}}}$

These converge for everyone . ${\displaystyle a>1}$

Be a function that holds true for ${\displaystyle f\colon \mathbb {N} \to \mathbb {R} }$

${\displaystyle f(t)\leq {\frac {c}{t^{\alpha }\cdot d(t!)-d((n-1)!)}}}$

where should be natural and constant; denote the number of divisors of . Then: ${\displaystyle t>0}$ ${\displaystyle \alpha >1,c>12}$ ${\displaystyle d(n)}$ ${\displaystyle n}$

${\displaystyle s_{12}(f)=\sum _{n=1}^{\infty }f(\mu (n))}$

is convergent.

Besides is also

${\displaystyle s_{13}=\sum _{n=1}^{\infty }{\frac {1}{\mu (1)!\cdot \mu (2)!\cdots \mu (n)!}}}$

convergent, as well as

${\displaystyle s_{14}(\alpha )=\sum _{n=1}^{\infty }{\frac {1}{\mu (n)!\cdot {\sqrt {\mu (n)!}}\cdot \log \left(\mu (n)\right)^{\alpha }}}}$

for . ${\displaystyle \alpha >1}$

Another convergent series is

${\displaystyle s_{15}=\sum _{n=1}^{\infty }{\frac {2^{n}}{\mu (2^{n})!}}.}$

Eventually also converges

${\displaystyle s_{16}(\alpha )=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{\alpha +1}}}}$

for everyone . ${\displaystyle \alpha >1}$

credentials

To give an overview

• Eric W. Weisstein : Smarandache Constants . In: MathWorld (English).
• Smarandache Function in PlanetMath
• Constant involving the Smarandache Function: http://fs.gallup.unm.edu//CONSTANT.TXT

Detailed works are

• I. Cojocaru, S. Cojocaru: The First Constant of Smarandache . in: Smarandache Notions Journal 7 (1996) (PDF; 5.4 MB) pp. 116–118.
• this. ibid. The Second Constant of Smarandache : pp. 119–120; and The Third and Fourth Constants of Smarandache : pp. 121-126.

• E. Burton: On Some Series Involving the Smarandache Function . In: Smarandache Function Journal 6 (1995) (PDF; 2.6 MB), pp. 13–15.

• E. Burton: On Some Convergent Series . In: Smarandache Notions Journal 7 (1996) (PDF; 5.4 MB): pp. 7–9.

• AJ Kempner: Miscellanea , in: The American Mathematical Monthly, Vol. 25, No. 5 (May 1918), pp. 201-210. jstor

• J. Sandor: On The Irrationality Of Certain Alternative Smarandache Series In: Smarandache Notions Journal 8 (1997) (PDF; 8.8 MB) pp. 143-144.