number theory, the prime omega functions ω ( n ) {\displaystyle \omega (n)} and Ω ( n ) {\displaystyle \Omega (n)} count the number of prime factors of a natural...

In mathematics, omega function refers to a function using the Greek letter omega, written ω or Ω. Ω {\displaystyle \Omega } (big omega) may refer to: The...

prime omega functions ω and Ω are defined by ω(n) = k, Ω(n) = a1 + a2 + ... + ak. To avoid repetition, whenever possible formulas for the functions listed...

mathematics and computer science: In complex analysis, the Omega constant, a solution of Lambert's W function In differential geometry, the space of differential...

function Ω(n), defined as the total number of prime factors of n, counting multiple factors multiple times, sometimes called the "Big Omega function"...

In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by Glaisher (1891). It is defined as the following infinite...

O notation Omega and agemo subgroup, in group theory Ω, an element in a subsumption lattice Prime omega function, in number theory OMEGA, an instrument...

) The prime omega functions count the number of primes, with (Ω) or without (ω) multiplicity: ω ( n ) = k , {\displaystyle \omega (n)=k,} Ω ( n ) = a...

number theory, may denote the prime omega function. That is, ω ( n ) {\displaystyle \omega (n)} is the number of distinct prime factors of the integer n....

multiplicity of a prime which does not divide n may be called 0 or may be considered undefined. Ω(n), the prime omega function, is the number of prime factors of...

OEIS) The Big Omega function is given by Ω ( x ) = the number of prime factors of  x  counted by multiplicities {\displaystyle \Omega (x)={\text{the...

Möbius μ function: Sum of the nth primitive roots of unity, it depends on the prime factorization of n. Prime omega functions Chebyshev functions Liouville...

Euler's method to solve the twin prime conjecture, that there exist infinitely many twin primes. The prime-counting function π ( n ) {\displaystyle \pi (n)}...

Liouville function, ω ( n ) {\displaystyle \omega (n)} is the number of distinct prime divisors of n {\displaystyle n} , and Ω ( n ) {\displaystyle \Omega (n)}...

This sieve is similar to the sieve of Eratosthenes that generates the primes, but it eliminates numbers based on their position in the remaining set...

{\displaystyle \sum _{k=0}^{22}\omega (n+k)\leq 57} , where ω ( n ) {\displaystyle \omega (n)} is the prime omega function for distinct prime factors. The corresponding...

{\displaystyle 2^{\omega (b)}} zeroes to g ( x ) = x 2 − x {\displaystyle g(x)=x^{2}-x} , where the prime omega function ω ( b ) {\displaystyle \omega (b)} is the...

an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem...

Mangoldt's function | μ | ∗ 1 = 2 ω , {\displaystyle |\mu |\ast 1=2^{\omega },} where ω ( n ) {\displaystyle \omega (n)} is the prime omega function counting...

multiplicative) f are given here for the prime omega functions ω ( n ) {\displaystyle \omega (n)} and Ω ( n ) {\displaystyle \Omega (n)} , which respectively count...

since 72 = 49, one has 6 × 8 = 48 {\displaystyle 6\times 8=48} . Since a prime number has factors of only 1 and itself, and since m = 2 is the only non-zero...

eventually reaches 1 when iterated over the perfect digital invariant function for p = 2 {\displaystyle p=2} . The origin of happy numbers is not clear...

generating function of the Fibonacci sequence, ∑ i = 0 ∞ F i z i {\displaystyle \sum _{i=0}^{\infty }F_{i}z^{i}} , is the rational function z 1 − z − z...

However, for prime numbers, the function also returns −1 and μ ( 1 ) = 1 {\displaystyle \mu (1)=1} . For a number n with one or more repeated prime factors...

In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse...

{1}{\beta }}\sum _{i\omega }g(i\omega )={\frac {1}{2\pi i\beta }}\oint g(z)h_{\eta }(z)\,dz,} As in Fig. 1, the weighting function generates poles (red...

\left\lfloor 2^{2^{\omega }}\right\rfloor ,\left\lfloor 2^{2^{2^{\omega }}}\right\rfloor ,\dots } are all prime. Let π(x) be the number of primes less than or...

the prime factorization of n (can be also seen as the sum of all the primes' exponents): Ω ( n ) := ∑ a i if n = ∏ p i a i . {\displaystyle \Omega (n):=\sum...

is an integer whose prime factors are all less than or equal to n. For example, a 7-smooth number is a number in which every prime factor is at most 7...

L5466311, with 1,142,392 decimal digits. If Ln is prime then n is 0, prime, or a power of 2. L2m is prime for m = 1, 2, 3, and 4 and no other known values...