104 (number)

From HandWiki
Short description: Natural number
← 103 104 105 →
Cardinalone hundred four
Ordinal104th
(one hundred fourth)
Factorization23 × 13
Divisors1, 2, 4, 8, 13, 26, 52, 104
Greek numeralΡΔ´
Roman numeralCIV
Binary11010002
Ternary102123
Quaternary12204
Quinary4045
Senary2526
Octal1508
Duodecimal8812
Hexadecimal6816
Vigesimal5420
Base 362W36

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

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

The Tits group [math]\displaystyle{ \mathbb {T} }[/math], 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 [math]\displaystyle{ \mathfrak{f}_4 }[/math] in 52 dimensions, whose associated lattice structure [math]\displaystyle{ \mathrm {F_{4}} }[/math] 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. "Sequence A006145 (Ruth-Aaron numbers (1): sum of prime divisors of n is equal to the sum of prime divisors of n+1.)". OEIS Foundation. https://oeis.org/A006145. Retrieved 2023-07-31. 
  2. ↑ Sloane, N. J. A., ed. "Sequence A033950 (Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.)". OEIS Foundation. https://oeis.org/A033950. Retrieved 2023-07-31. 
  3. ↑ Sloane, N. J. A., ed. "Sequence A006036 (Primitive pseudoperfect numbers.)". OEIS Foundation. https://oeis.org/A006036. Retrieved 2016-05-27. 
  4. ↑ Sloane, N. J. A., ed. "Sequence A000217 (Triangular numbers)". OEIS Foundation. https://oeis.org/A000217. Retrieved 2023-07-31. 
  5. ↑ Sloane, N. J. A., ed. "Sequence A002110 (Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.)". OEIS Foundation. https://oeis.org/A002110. Retrieved 2023-07-31. 
  6. ↑ Sloane, N. J. A., ed. "Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and prime to n.)". OEIS Foundation. https://oeis.org/A000010. Retrieved 2023-07-31. 
  7. ↑ Sloane, N. J. A., ed. "Sequence A006872 (Numbers k such that phi(k) is phi(sigma(k)))". OEIS Foundation. https://oeis.org/A006872. 
  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. 
  9. ↑ 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.)". OEIS Foundation. https://oeis.org/A306302. Retrieved 2022-05-09. 
  10. ↑ 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.)". OEIS Foundation. https://oeis.org/A007267. 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. https://www.tandfonline.com/doi/abs/10.1081/AGB-100002175. 
  • Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 133



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