The Narayana numbers are a sequence of numbers that appear in combinatorial mathematics and are related to various counting problems, including those involving paths and combinations.
The noncototient is a mathematical concept related to number theory. Specifically, it refers to the integers \( n \) for which the equation \( \phi(m) = n \) has no solution for any integer \( m \). Here, \( \phi(m) \) is the Euler's totient function, which counts the number of positive integers up to \( m \) that are relatively prime to \( m \).
A nonhypotenuse number is not a standard term in mathematics, so its meaning may vary depending on context. However, it could be inferred as a number that cannot be the length of the hypotenuse of a right triangle, based on the properties of right triangles in Euclidean geometry.
A **nontotient** is a positive integer \( n \) for which there is no integer \( k \) such that \( k \) and \( n \) are coprime, and \( \phi(k) = n \), where \( \phi \) is the Euler's totient function. The Euler's totient function \( \phi(k) \) counts the number of integers up to \( k \) that are coprime to \( k \).
An odious number is a non-negative integer that has an odd number of 1s in its binary representation. In contrast, a number that has an even number of 1s in its binary form is referred to as an "elegant number." For example: - The number 3 in binary is `11`, which contains two 1s (an even number), so it is not odious.
The On-Line Encyclopedia of Integer Sequences (OEIS) is a comprehensive database that collects and catalogs integer sequences. Launched in 1964 by Neil J. A. Sloane, the OEIS has grown significantly over the years and is now a valuable resource for mathematicians, scientists, and hobbyists interested in number theory, combinatorics, and other areas involving sequences of integers.
The Ordered Bell number is a concept in combinatorial mathematics that counts the number of ways to partition a set into a certain number of non-empty ordered subsets. More formally, the \( n \)-th Ordered Bell number, denoted as \( B_n^{o} \), gives the number of ways to partition a set of size \( n \) into \( k \) non-empty subsets, where the order of the subsets matters.
In number theory and combinatorics, the **partition function** is a function that counts the number of distinct ways a given positive integer can be expressed as a sum of positive integers, regardless of the order of addends.
Pell numbers are a sequence of integers defined by a specific recurrence relation. The Pell numbers are similar to the Fibonacci numbers but are defined differently. The sequence starts with initial values, and each subsequent number is derived from the previous two.
A **perfect power** is a positive integer that can be expressed in the form \( n = a^k \), where \( a \) is a positive integer and \( k \) is an integer greater than 1. In other words, a number is a perfect power if it can be represented as an integer raised to an integer power greater than one. For example: - \( 4 \) is a perfect power because \( 4 = 2^2 \).
Perrin numbers are a sequence of numbers defined by a specific recurrence relation, similar in nature to the Fibonacci sequence.
A Pillai sequence is a specific type of integer sequence defined in number theory. It is named after the Indian mathematician S. P. Pillai. The sequence is generated using a recurrence relation based on the properties of prime numbers.
The Poly-Bernoulli numbers, denoted as \( B_{n}^{(k)} \), are a generalization of the classical Bernoulli numbers. They are defined in the context of polyadic and combinatorial number theory, particularly in relation to the study of special sequences and functions.
A **powerful number** is a positive integer \( n \) such that for every prime \( p \) that divides \( n \), \( p^2 \) also divides \( n \). In other words, if a prime number appears in the factorization of a powerful number, it must appear with an exponent of at least 2.
A practical number is a positive integer \( n \) that can be represented as a sum of distinct positive integers not exceeding \( n \). In other words, for a number to be practical, any integer up to \( n \) can be expressed as a sum of distinct integers chosen from the set of positive integers less than or equal to \( n \).
A primary pseudoperfect number is a type of integer closely related to the concepts of number theory, particularly with respect to the properties of its divisors. A positive integer \( n \) is called a primary pseudoperfect number if it can be expressed as the sum of a subset of its proper divisors (the divisors excluding itself) plus one.