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Golden ratio base is a non-integer positional numeral system that uses the golden ratio as its base. It is sometimes referred to as base-φ, golden mean base, phi-base, or, colloquially, phinary. Any non-negative real number can be represented as a base-φ numeral using only the digits 0 and 1, and avoiding the digit sequence "11" – this is called a standard form. A base-φ numeral that includes the digit sequence "11" can always be rewritten in standard form, using the algebraic properties of the base φ — most notably that φ1 + φ0 = φ2. For instance, 11φ = 100φ.
222 is the natural number following 221 and preceding 223.
In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy.
72 (seventy-two) is the natural number following 71 and preceding 73. It is half a gross or 6 dozen.
36 (thirty-six) is the natural number following 35 and preceding 37.
The Lucas sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci sequence. Individual numbers in the Lucas sequence are known as Lucas numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.
48 (forty-eight) is the natural number following 47 and preceding 49. It is one third of a gross, or four dozens.
120 is the natural number following 119 and preceding 121.
360 is the natural number following 359 and preceding 361.
144 is the natural number following 143 and preceding 145. It is a dozen dozens, or one gross.
720 is the natural number following 719 and preceding 721.
In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1, all of whose Galois conjugates are less than 1 in absolute value. These numbers were discovered by Axel Thue in 1912 and rediscovered by G. H. Hardy in 1919 within the context of diophantine approximation. They became widely known after the publication of Charles Pisot's dissertation in 1938. They also occur in the uniqueness problem for Fourier series. Tirukkannapuram Vijayaraghavan and Raphael Salem continued their study in the 1940s. Salem numbers are a closely related set of numbers.
In number theory, a nontotient is a positive integer n which is not a totient number: it is not in the range of Euler's totient function φ, that is, the equation φ(x) = n has no solution x. In other words, n is a nontotient if there is no integer x that has exactly n coprimes below it. All odd numbers are nontotients, except 1, since it has the solutions x = 1 and x = 2. The first few even nontotients are
In number theory, a noncototient is a positive integer n that cannot be expressed as the difference between a positive integer m and the number of coprime integers below it. That is, m − φ(m) = n, where φ stands for Euler's totient function, has no solution for m. The cototient of n is defined as n − φ(n), so a noncototient is a number that is never a cototient.
A highly totient number is an integer that has more solutions to the equation , where is Euler's totient function, than any integer below it. The first few highly totient numbers are
In number theory, a branch of mathematics, a highly cototient number is a positive integer which is above 1 and has more solutions to the equation
225 is the natural number following 224 and preceding 226.
In mathematics, the Golomb sequence, named after Solomon W. Golomb, is a monotonically increasing integer sequence where an is the number of times that n occurs in the sequence, starting with a1 = 1, and with the property that for n > 1 each an is the smallest unique integer which makes it possible to satisfy the condition. For example, a1 = 1 says that 1 only occurs once in the sequence, so a2 cannot be 1 too, but it can be 2, and therefore must be 2. The first few values are
60,000 (sixty thousand) is the natural number that comes after 59,999 and before 60,001. It is a round number. It is the value of (F25).
In mathematics, the Wythoff array is an infinite matrix of integers derived from the Fibonacci sequence and named after Dutch mathematician Willem Abraham Wythoff. Every positive integer occurs exactly once in the array, and every integer sequence defined by the Fibonacci recurrence can be derived by shifting a row of the array.
References
- ↑ Sloane, N. J. A. (ed.). "SequenceA006872". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.