104 (number)

← 103 104 105 →
← 100 101 102 103 104 105 106 107 108 109 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	one hundred four
Ordinal	104th (one hundred fourth)
Factorization	23 × 13
Divisors	1, 2, 4, 8, 13, 26, 52, 104
Greek numeral	ΡΔ´
Roman numeral	CIV
Binary	11010002
Ternary	102123
Senary	2526
Octal	1508
Duodecimal	8812
Hexadecimal	6816

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 ${\displaystyle \mathbb {B} }$, its McKay–Thompson series representative of a principal modular function is ${\displaystyle T_{2A}(\tau )}$, with constant term ${\displaystyle a(0)=104}$: ^[10]

${\displaystyle j_{2A}(\tau )=T_{2A}(\tau )+104={\frac {1}{q}}+104+4372q+96256q^{2}+\cdots }$

The Tits group ${\displaystyle \mathbb {T} }$, 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 ${\displaystyle {\mathfrak {f}}_{4}}$ in 52 dimensions, whose associated lattice structure ${\displaystyle \mathrm {F_{4}} }$ 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:

• The atomic number of rutherfordium.
• The number of Corinthian columns in the Temple of Olympian Zeus, the largest temple ever built in Greece.
• The number of Symphonies written by Joseph Haydn upon which numbers are agreed (though in fact, he wrote two more: see list of symphonies by Joseph Haydn).

See also

• List of highways numbered 104
• The years 104 BC and AD 104.

References

1. Sloane, N. J. A. (ed.). "SequenceA006145(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.
2. Sloane, N. J. A. (ed.). "SequenceA033950(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.
3. Sloane, N. J. A. (ed.). "SequenceA006036(Primitive pseudoperfect numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-27.
4. Sloane, N. J. A. (ed.). "SequenceA000217(Triangular numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-31.
5. Sloane, N. J. A. (ed.). "SequenceA002110(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.
6. Sloane, N. J. A. (ed.). "SequenceA000010(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.
7. Sloane, N. J. A. (ed.). "SequenceA006872(Numbers k such that phi(k) is phi(sigma(k)))". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
8. Winkler, Mike; Dinkelacker, Peter; Vogel, Stefan (2017). "New minimal (4; n)-regular matchstick graphs". Geombinatorics Quarterly. Colorado Springs, CO: University of Colorado, Colorado Springs. XXVII (1): 26–44. arXiv: 1604.07134 . S2CID   119161796. Zbl   1373.05125.
9. Sloane, N. J. A. (ed.). "SequenceA306302(...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.
10. Sloane, N. J. A. (ed.). "SequenceA007267(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.
11. Lubeck, Frank (2001). "Smallest degrees of representations of exceptional groups of Lie type". Communications in Algebra . Philadelphia, PA: Taylor & Francis. 29 (5): 2151. doi:10.1081/AGB-100002175. MR   1837968. S2CID   122060727. Zbl   1004.20003.

• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 133