def divisor_sum_using_prime_factors(n):
pf = list(set(prime_factor.prime_factors(n)))
s = 1
for p in pf:
k = 0
while n % p == 0:
k += 1
n //= p
s *= (p**(k + 1) - 1) // (p - 1)
return s
def factorise(n):
if n < 1:
return None
factors = [1]
pfactors = [x for x in prime_factors(n)]
pfactors = [x for x in chain.from_iterable(combinations(pfactors, r) for r in range(len(pfactors) + 1))]
for comb in pfactors:
product = 1
for x in comb:
product *= x
if product not in factors:
factors.append(product)
return sorted(factors)
def totient(num):
"""
Counts the numbers that are relative prime to the number.
This means that if a number 'x' has common divisors with another number 'a' lower than itself,
it is, 'a' is not relative prime to 'x'.
This means that if a number is prime, its totient function is its value - 1, as
all numbers lower than the number itself is relative prime to the number.
An example: totient(9):
1, 2, 4, 5, 7, 8 are all relative prime
3, 6, and 9 have at least one common divisor with 9
returns 6, as there are 6 numbers relative prime to 9
If you visit 'https://en.wikipedia.org/wiki/Euler%27s_totient_function'
this function uses the function described under euler's product formula.
"""
# set(prime_factors(num)) to get only unique primes.
# it is the (1 - 1/p) part from the wiki-page
uniqe_primes = [(1 - 1 / p) for p in set(prime_factors(num))]
phi = num * product(
uniqe_primes) # The totient function is commonly called phi
phi = int(round(phi)) # As phi is a whole number
return phi
def phi(n): #Eulers phi-funktion
#Giver p-1 for alle primfaktorerne
return produkt([p - 1 for p in prime_factors(n)]) #Og ganger dem så sammen
def test_prime(self): self.assertEqual(prime_factors(5), [5,])
def test_known(self): self.assertEqual(prime_factors(13195), [5, 7, 13, 29,])
def test_nonprime_noncomposite(self): self.assertFalse(prime_factors(1))
def test_composite(self): self.assertEqual(prime_factors(6), [2, 3,])