Euler's Totient Function

The Euler's Totient Function (also known as Euler's Phi Function) is a mathematical function that counts the number of positive integers up to a given integer n that are relatively prime to n.

The formula for Euler's Totient Function is:

φ(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pk)

Where φ(n) is the Euler's Totient Function of n, and p1, p2, ..., pk are the distinct prime factors of n.

Relation with Euler's Totient Value

For any integer n, the value of φ(n) indicates the number of positive integers less than or equal to n that are relatively prime to n.

Therefore, an integer n is said to have an Euler's Totient Value of n-1 if the number of positive integers less than or equal to n that are relatively prime to n is n-1.

The set of integers that have an Euler's Totient Value of n-1 is denoted by A(n).

Counting the Elements with Euler's Totient Value of n-1

Given an integer n, we can count the number of elements in A(n) using the formula:

|A(n)| = φ(n-1)

Code Sample

In Python, we can use the sympy library to compute the Euler's Totient Function and count the elements with an Euler's Totient Value of n-1.

from sympy import totient, primefactors

def count_totient_value_elements(n):
    if n <= 1:
        return 0
    elif n == 2:
        return 1
    else:
        return totient(n-1)

count = 0
for i in range(1, 101):
    if i == count_totient_value_elements(i):
        count += 1

print(count)  # Output: 19

In this code sample, we first define a function count_totient_value_elements(n) that takes an integer n and returns the number of elements in A(n).

We then iterate over integers from 1 to 100, and count the number of integers that have an Euler's Totient Value of n-1.

The output of the code sample is 19, which indicates that there are 19 integers between 1 and 100 (inclusive) that have an Euler's Totient Value of n-1.