def euler():
# number of fractions found
match_count = 0
# for each possible numerator n
for n in range(2, math.ceil(MAX / 2) + 1):
# lowest possible denominator for a fraction in the wanted range
start = math.ceil(2 * n + DELTA)
# highest possible denominator for a fraction in the wanted range
end = min(MAX, int(3 * n - DELTA))
# construct a dictionary whose keys are all the denominators that
# do *not* make a reduced fraction with the numerator n
table = dict()
for p in prime.prime_factors(n):
for q in range(p, end + 1, p):
table[q] = True
# for each denominator d
for d in range(start, end + 1):
# if n / d is a reduced fraction
if not d in table:
match_count += 1
# return the number of fractions found
return match_count
def totient(n):
"""Return the number of integers <= n that are relatively prime with n."""
t = n
for p in set(prime.prime_factors(n)):
t *= 1 - 1 / p
return round(t)
