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.
|Date of creation||2013-03-22 13:30:16|
|Last modified on||2013-03-22 13:30:16|
|Last modified by||Wkbj79 (1863)|