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
p
n
{\ displaystyle p_ {n}}
n
{\ displaystyle n}
n
{\ displaystyle n}
p
n
+
1
-
p
n
<
1.
{\ displaystyle {\ sqrt {p_ {n + 1}}} - {\ sqrt {p_ {n}}} <1.}
This assumption can be generalized as follows:
p
n
+
1
a
-
p
n
a
<
1
f
u
¨
r
a
l
l
e
a
<
a
0
.
{\ displaystyle p_ {n + 1} ^ {a} -p_ {n} ^ {a} <1 \ qquad \ quad \ mathrm {f {\ ddot {u}} r \; all} \; a <a_ { 0}.}
This upper limit for , approximately , is often referred to as
the Smarandache constant. is solution of the equation .
a
0
{\ displaystyle a_ {0}}
0.567
14813 ...
{\ displaystyle 0 {,} 56714813 ...}
a
0
{\ displaystyle a_ {0}}
p
31
x
-
p
30th
x
=
127
x
-
113
x
=
1
{\ displaystyle p_ {31} ^ {x} -p_ {30} ^ {x} = 127 ^ {x} -113 ^ {x} = 1}
2.
The Smarandache function is defined as follows:
μ
(
n
)
{\ displaystyle \ mu (n)}
μ
(
n
)
{\ displaystyle \ mu (n)}
is the smallest natural number that is divisible by .
μ
(
n
)
!
{\ displaystyle \ mu (n)!}
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.
μ
(
8th
)
{\ displaystyle \ mu (8)}
μ
(
8th
)
=
4th
{\ displaystyle \ mu (8) = 4}
Smarandache constants
The first Smarandache constant is defined by
s
1
=
∑
n
=
2
∞
1
μ
(
n
)
!
=
1.093
170459 ...
{\ 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: .
μ
(
n
)
≤
n
{\ displaystyle \ mu (n) \ leq n}
∑
1
μ
(
n
)
!
<
∑
1
n
!
=
e
{\ 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
s
2
=
∑
n
=
2
∞
μ
(
n
)
n
!
=
1.714
0062935916 ...
{\ 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
s
3
=
∑
n
=
2
∞
1
μ
(
2
)
⋅
μ
(
3
)
⋯
μ
(
n
)
=
0.719
9607000437 ...
{\ 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 :
α
≥
1
{\ displaystyle \ alpha \ geq 1}
s
4th
(
α
)
=
∑
n
=
2
∞
n
α
μ
(
2
)
⋅
μ
(
3
)
⋯
μ
(
n
)
{\ displaystyle s_ {4} (\ alpha) = \ sum _ {n = 2} ^ {\ infty} {\ frac {n ^ {\ alpha}} {\ mu (2) \ cdot \ mu (3) \ cdots \ mu (n)}}}
The first values for natural :
α
{\ displaystyle \ alpha}
α
{\ displaystyle \ alpha}
S.
4th
(
α
)
{\ 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
s
5
=
∑
n
=
1
∞
(
-
1
)
n
-
1
μ
(
n
)
n
!
{\ displaystyle s_ {5} = \ sum _ {n = 1} ^ {\ infty} {\ frac {(-1) ^ {n-1} \ mu (n)} {n!}}}
also has a limit. The next constant
s
6th
=
∑
n
=
2
∞
μ
(
n
)
(
n
+
1
)
!
,
{\ displaystyle s_ {6} = \ sum _ {n = 2} ^ {\ infty} {\ frac {\ mu (n)} {(n + 1)!}},}
converges to a value .
0.218
282
<
s
6th
<
0
,
5
{\ displaystyle 0 {,} 218282 <s_ {6} <0 {,} 5}
More generally, they even converge
s
7th
=
∑
n
=
2
∞
μ
(
n
)
(
n
+
k
)
!
and
s
8th
=
∑
n
=
2
∞
μ
(
n
)
(
n
-
k
)
!
{\ displaystyle s_ {7} = \ sum _ {n = 2} ^ {\ infty} {\ frac {\ mu (n)} {(n + k)!}} \ quad {\ text {and}} \ quad s_ {8} = \ sum _ {n = 2} ^ {\ infty} {\ frac {\ mu (n)} {(nk)!}}}
for natural (or whole ) .
k
≠
0
{\ displaystyle k \ not = 0}
Also converges
s
9
=
∑
n
=
2
∞
1
μ
(
2
)
2
!
+
μ
(
3
)
3
!
+
⋯
+
μ
(
n
)
n
!
.
{\ 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
s
10
(
α
)
=
∑
n
=
2
∞
1
μ
(
n
)
α
μ
(
n
)
!
{\ displaystyle s_ {10} (\ alpha) = \ sum _ {n = 2} ^ {\ infty} {\ frac {1} {\ mu (n) ^ {\ alpha} {\ sqrt {\ mu (n )!}}}}}
and
s
11
(
α
)
=
∑
n
=
2
∞
1
μ
(
n
)
α
(
μ
(
n
)
+
1
)
!
{\ displaystyle s_ {11} (\ alpha) = \ sum _ {n = 2} ^ {\ infty} {\ frac {1} {\ mu (n) ^ {\ alpha} {\ sqrt {(\ mu ( n) +1)!}}}}}
These converge for everyone .
a
>
1
{\ displaystyle a> 1}
Be a function that holds true for
f
:
N
→
R.
{\ displaystyle f \ colon \ mathbb {N} \ to \ mathbb {R}}
f
(
t
)
≤
c
t
α
⋅
d
(
t
!
)
-
d
(
(
n
-
1
)
!
)
{\ 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:
t
>
0
{\ displaystyle t> 0}
α
>
1
,
c
>
12
{\ displaystyle \ alpha> 1, c> 12}
d
(
n
)
{\ displaystyle d (n)}
n
{\ displaystyle n}
s
12
(
f
)
=
∑
n
=
1
∞
f
(
μ
(
n
)
)
{\ displaystyle s_ {12} (f) = \ sum _ {n = 1} ^ {\ infty} f (\ mu (n))}
is convergent.
Besides is also
s
13
=
∑
n
=
1
∞
1
μ
(
1
)
!
⋅
μ
(
2
)
!
⋯
μ
(
n
)
!
{\ displaystyle s_ {13} = \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {\ mu (1)! \ cdot \ mu (2)! \ cdots \ mu (n)! }}}
convergent, as well as
s
14th
(
α
)
=
∑
n
=
1
∞
1
μ
(
n
)
!
⋅
μ
(
n
)
!
⋅
log
(
μ
(
n
)
)
α
{\ displaystyle s_ {14} (\ alpha) = \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {\ mu (n)! \ cdot {\ sqrt {\ mu (n)! }} \ cdot \ log \ left (\ mu (n) \ right) ^ {\ alpha}}}}
for .
α
>
1
{\ displaystyle \ alpha> 1}
Another convergent series is
s
15th
=
∑
n
=
1
∞
2
n
μ
(
2
n
)
!
.
{\ displaystyle s_ {15} = \ sum _ {n = 1} ^ {\ infty} {\ frac {2 ^ {n}} {\ mu (2 ^ {n})!}}.}
Eventually also converges
s
16
(
α
)
=
∑
n
=
1
∞
μ
(
n
)
n
α
+
1
{\ displaystyle s_ {16} (\ alpha) = \ sum _ {n = 1} ^ {\ infty} {\ frac {\ mu (n)} {n ^ {\ alpha +1}}}}
for everyone .
α
>
1
{\ displaystyle \ alpha> 1}
credentials
To give an overview
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.
AJ Kempner: Miscellanea , in: The American Mathematical Monthly, Vol. 25, No. 5 (May 1918), pp. 201-210. jstor
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