function
The function, also called the divisor function, takes a positive integer as its input and gives the number of positive divisors of its input as its output. For example, since , , and are all of the positive divisors of , we have . As another example, since , , , and are all of the positive divisors of , we have .
The function behaves according to the following two rules:
1. If is a prime and is a nonnegative integer, then .
2. If , then .
Because these two rules hold for the function, it is a multiplicative function.
Note that these rules work for the previous two examples. Since is prime, we have . Since and are distinct primes, we have .
If is a positive integer, the number of prime factors (http://planetmath.org/UFD) of over is . For example, and .
The function is extremely useful for studying cyclic rings.
The sequence appears in the OEIS as sequence http://www.research.att.com/ njas/sequences/A000005A000005.
Title | function |
---|---|
Canonical name | tauFunction |
Date of creation | 2013-03-22 13:30:16 |
Last modified on | 2013-03-22 13:30:16 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 21 |
Author | Wkbj79 (1863) |
Entry type | Definition |
Classification | msc 11A25 |
Synonym | divisor function |
Related topic | Divisor |
Related topic | DirichletHyperbolaMethod |
Related topic | 2omeganLeTaunLe2Omegan |
Related topic | Divisibility |
Related topic | ValuesOfNForWhichVarphintaun |
Related topic | LambertSeries |
Related topic | ParityOfTauFunction |