From Wikipedia, the free encyclopedia
Natural number
104 (one hundred [and] four ) is the
natural number following
103 and preceding
105 .
In mathematics
104 forms the fifth
Ruth-Aaron pair with
105 , since the distinct prime factors of 104 (
2 and
13 ) and 105 (
3 ,
5 , and
7 ) both add up to
15 .
[1] Also, the sum of the
divisors of 104 aside from
unitary divisors , is 105. With eight total
divisors where
8 is the fourth largest, 104 is the seventeenth
refactorable number .
[2] 104 is also the twenty-fifth
primitive semiperfect number .
[3]
The
sum of all its divisors is σ (104) =
210 , which is the sum of the first twenty nonzero
integers ,
[4] as well as the product of the first four prime numbers (2 × 3 × 5 × 7).
[5]
Its
Euler totient , or the number of integers
relatively prime with 104, is
48 .
[6] This value is also equal to the totient of its sum of divisors, φ (104) = φ (σ (104)).
[7]
The smallest known 4-regular
matchstick graph has 104
edges and 52
vertices , where four unit
line segments intersect at every vertex.
[8]
A row of four adjacent
congruent
rectangles can be divided into a maximum of
104 regions , when extending
diagonals of all possible rectangles.
[9]
Regarding the
second largest sporadic group
B
{\displaystyle \mathbb {B} }
, its
McKay–Thompson series representative of a
principal modular function is
T
2
A
(
τ
)
{\displaystyle T_{2A}(\tau )}
, with constant term
a
(
0
)
=
104
{\displaystyle a(0)=104}
:
[10]
j
2
A
(
τ
)
=
T
2
A
(
τ
)
+
104
=
1
q
+
104
+
4372
q
+
96256
q
2
+
⋯
{\displaystyle j_{2A}(\tau )=T_{2A}(\tau )+104={\frac {1}{q}}+104+4372q+96256q^{2}+\cdots }
The
Tits group
T
{\displaystyle \mathbb {T} }
, which is the only
finite simple group to classify as either a non-strict group of
Lie type or
sporadic group , holds a
minimal faithful complex representation in 104 dimensions.
[11] This is twice the dimensional representation of
exceptional Lie algebra
f
4
{\displaystyle {\mathfrak {f}}_{4}}
in 52 dimensions, whose associated
lattice structure
F
4
{\displaystyle \mathrm {F_{4}} }
forms the
ring of
Hurwitz quaternions that is represented by the vertices of the
24-cell — with this
regular 4-polytope one of 104 total four-dimensional
uniform polychora , without taking into account the infinite families of
uniform antiprismatic prisms and
duoprisms .
In other fields
104 is also:
See also
References
^
Sloane, N. J. A. (ed.).
"Sequence A006145 (Ruth-Aaron numbers (1): sum of prime divisors of n is equal to the sum of prime divisors of n+1.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-31 .
^
Sloane, N. J. A. (ed.).
"Sequence A033950 (Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-31 .
^
Sloane, N. J. A. (ed.).
"Sequence A006036 (Primitive pseudoperfect numbers.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-27 .
^
Sloane, N. J. A. (ed.).
"Sequence A000217 (Triangular numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-31 .
^
Sloane, N. J. A. (ed.).
"Sequence A002110 (Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-31 .
^
Sloane, N. J. A. (ed.).
"Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and prime to n.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-31 .
^
Sloane, N. J. A. (ed.).
"Sequence A006872 (Numbers k such that phi(k) is phi(sigma(k)))" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^ Winkler, Mike; Dinkelacker, Peter; Vogel, Stefan (2017). "New minimal (4; n)-regular matchstick graphs". Geombinatorics Quarterly . XXVII (1). Colorado Springs, CO:
University of Colorado, Colorado Springs : 26–44.
arXiv :
1604.07134 .
S2CID
119161796 .
Zbl
1373.05125 .
^
Sloane, N. J. A. (ed.).
"Sequence A306302 (...Number of regions (or cells) formed by drawing the line segments connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-05-09 .
^
Sloane, N. J. A. (ed.).
"Sequence A007267 (Expansion of 16 * (1 + k^2)^4 /(k * k'^2)^2 in powers of q where k is the Jacobian elliptic modulus, k' the complementary modulus and q is the nome.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-31 .
^ Lubeck, Frank (2001).
"Smallest degrees of representations of exceptional groups of Lie type" .
Communications in Algebra . 29 (5). Philadelphia, PA:
Taylor & Francis : 2151.
doi :
10.1081/AGB-100002175 .
MR
1837968 .
S2CID
122060727 .
Zbl
1004.20003 .
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