The Bell series of this function is equal to one minus x over 1 minus px, while the partial sums of this function show that its average order is equal to 3n divided by pi squared, a result which can be used to find the number of lattice points visible from the origin. The Dirichlet convolution of the identity function N and the Mobius function gives this function, and if a number has r distinct odd prime factors, 2 to the r divides this function of the number. Summing this function over the divisors of a number n gives n. It gives the number of elements in the group Z 
The Bell series of this function is equal to one minus x over 1 minus px, while the partial sums of this function show that its average order is equal to 3n divided by pi squared, a result which can be used to find the number of lattice points visible from the origin. The Dirichlet convolution of the identity function N and the Mobius function gives this function, and if a number has r distinct odd prime factors, 2 to the r divides this function of the number. Summing this function over the divisors of a number n gives n. It gives the number of elements in the group Z sub n star, and for a prime number p this function returns p minus 1. Also used in its namesake's generalization of Fermat's Little Theorem, for 10 points, name this function that returns the number of integers less than and relatively prime to the input.
The Bell series of this function is equal to one minus x over 1 minus px, while the partial sums of this function show that its average order is equal to 3n divided by pi squared, a result which can be used to find the number of lattice points visible from the origin.
Euler phi function [or Euler totient function]