Common product functions (from encyclopedia)

We used the over-product function for hdu1452... I met again just now. So remind yourself of some information.

In the field of Non-number theory, product functions refer to all functions that have properties F (AB) = f (a) F (B) for any A and B.

Product Function in number theory: For an arithmetic function f (n) of positive integer N, if F (1) = 1, and when a and B are mutually Prime, F (AB) = f (a) F (B) is called as a product function in number theory.

If F (n) is a product function, F (AB) = f (a) F (B ), it is fully product. [1]

S (6) = S (2) * s (3) = 3*4 = 12;

S (20) = S (4) * s (5) = 7*6 = 42;

Look at S (50) = 1 + 2 + 5 + 10 + 25 + 50 = 93 = 3*31 = S (2) * s (25), S (25) = 1 + 5 + 25 = 31.

This is called a product function in number theory. When gcd (a, B) = 1, S (A * B) = S (a) * s (B );

Nature 1

The value of the product function is completely determined by the power of the prime number, which is related to the basic arithmetic theorem.

That is to say, if n is expressed as a quality factor factorization

 

Then there is

 

Nature 2

If F is a product function and has

 

Then f is a fully product function.

Product

Number of positive integers in the interaction with N using the (N)-Euler function

μ (N)-Mobius function, about the number of quality factors in the non-Gini number

Gcd (n, k)-the most common factor, when K is fixed

Number of positive factors in D (N)-N

Sum of all positive factors of σ (N)-N

σ k (N)-factor function, the sum of K power of all positive factors of n, where K can be any plural.

1 (n)-constant function, defined as 1 (n) = 1 (full product)

ID (N)-unit function, defined as ID (n) = N (full product)

IDK (N)-power function. For any complex number or real number K, it is defined as IDK (n) = n ^ K (full product)

ε (N)-is defined as: If n = 1, ε (n) = 1; if n> 1, ε (n) = 0. It is not called "Multiplication unit for Dirichlet convolution" (full product)

λ (N)-Liu weier function, number of quality factors that can divide n

Gamma (N), defined as gamma (n) = (-1) ^ ω (N). Here, the addition function ω (n) is the number of classes with different divisible N values.

In addition, all Dirichlet features are fully product [1]

Non-Product

Von mangort: WHEN n is the integer power of prime number P, Lambda (n) = ln (p); otherwise, Lambda (n) = 0

Number of prime numbers not greater than positive integer N π (N)

Number of integer splits P (n): The number of methods in which an integer can represent the sum of positive integers [2]

Common product functions (from encyclopedia)