human
I am working on the file "P3wNT Notebook 7.ipynb".
The first task is:
The totient is a multiplicative function, meaning that if $GCD(a,b) = 1$, then $\phi(ab) = \phi(a) \phi(b)$.  Therefore, the totient of number can be found quickly from the totient of the prime powers within its decomposition.  We can re-implement the totient using our functions for prime decomposition and multiplicative functions from Notebook 4.
Can you write Python program for it?
 Give me a the Python program you requested?
assistant
#Here is the Python program you requested.

from math import sqrt  # We'll want to use the square root.

def smallest_factor(n):
    '''
    Gives the smallest prime factor of n.
    '''
    if n < 2:
        return None # No prime factors!
    
    test_factor = 2 # The smallest possible prime factor.
    max_factor = sqrt(n) # we don't have to search past sqrt(n).
    
    while test_factor <= max_factor:
        if n%test_factor == 0:
            return test_factor
        test_factor = test_factor + 1  # This could be sped up.
    
    return n  # If we didn't find a factor up to sqrt(n), n itself is prime!

def decompose(N):
    '''
    Gives the unique prime decomposition of a positive integer N,
    as a dictionary with primes as keys and exponents as values.
    '''
    current_number = N  # We'll divide out factors from current_number until we get 1.
    decomp = {} # An empty dictionary to start.
    while current_number > 1:
        p = smallest_factor(current_number) # The smallest prime factor of the current number.
        if p in decomp.keys():  # Is p already in the list of keys?
            decomp[p] = decomp[p] + 1 # Increase the exponent (value with key p) by 1.
        else:  # "else" here means "if p is not in decomp.keys()".
            decomp[p] = 1  # Creates a new entry in the dictionary, with key p and value 1.
        current_number = current_number // p # Factor out p.
    return decomp

def mult_function(f_pp):
    '''
    When a function f_pp(p,e) of two arguments is input,
    this outputs a multiplicative function obtained from f_pp
    via prime decomposition.
    '''
    def f(n):
        D = decompose(n)
        result = 1
        for p in D:
            result = result * f_pp(p, D[p])
        return result
    
    return f