King's gross

The king's gross is equal to 223.^[1] The term was coined by Garrett Wilkinson.

Properties[]

223 is deficient.^[8]
223 is the only nonnegative integer which cannot be written as a sum of 36 nonnegative fifth powers (see Waring's problem). It is also the largest integer which cannot be written as a sum of 32, 33, 34 or 35 nonnegative fifth powers; there are only fifteen, ten, six and three nonnegative integers with this property, respectively.
In some countries, such as China, the Band III ends at 223 MHz.
It is also the number of non-control 8-bit characters.
It is the successor of 222 and the preceder of 224.

Notation	Lower bound	Upper bound
Up-arrow notation	15 ↑ 2
Scientific notation	2.23 x 10²	2.231 x 10²
Slow-growing hierarchy	\(g_{\omega^{\omega}}(3)\)	\(g_{\omega^{\omega}}(4)\)
Copy notation	2[3]	3[3]
Chained arrow notation	15 → 2
Bowers' Exploding Array Function/Bird's array notation	{15,2}
Fast-growing hierarchy	f₂(5)	f₂(6)
Hardy hierarchy	H_ω(111)	H_ω(112)
Middle-growing hierarchy	m(ω,7)	m(ω,8)
Hyper-E notation	E2.3483
Hyper-E notation (non-10 base)	\(E[15]2\)
Hyperfactorial array notation	5!	6!
X-Sequence Hyper-Exponential Notation	15{1}2
Steinhaus-Moser Notation	3[3]	4[3]
PlantStar's Debut Notation	[1]	[2]
H* function	H(-0.3)	H(-0.2)
Bashicu matrix system with respect to version 4	(0)[14]	(0)[15]
m(n) map	m(1)(3)	m(1)(4)
s(n) map	\(s(1)(\lambda x . x+1)(110)\)	\(s(1)(\lambda x . x+1)(111)\)