def chain_length(n):
result = 2
n -= 1
while n > 1 and result < 26: # no point in computing chains longer than 25
n = totient(n)
result += 1
return result
def solve_slow():
result = {}
for n in xrange(2, 10000000):
phi = totient(n)
if is_permutation(n, phi):
result[n] = float(n) / phi
return min(result.iteritems(), key=operator.itemgetter(1))[0]
def solve():
d = 1
primes = []
phi = None
while True:
for p in gen_primes():
d *= p
primes.append(p)
phi = totient(d)
if phi / float(d-1) < LIMIT:
if p > 2:
primes.pop()
d /= p
phi = totient(d)
break
else:
return d
def solve():
primes = list_primes(1000000)
result = {}
for x in primes[:300:-1]:
for y in primes[300:]:
n = x * y
if n > 10000000:
break
phi = totient(n)
if is_permutation(n, phi):
result[n] = float(n) / phi
return min(result.iteritems(), key=operator.itemgetter(1))[0]
def maximum_totient_ratio_simple(max_n):
"""Return number n < max_n with maximum n/totient(n) ratio.
Per analysis, the n with maximum n/totient(n) for any given range will be an even number.
:param max_n: maximum n to check
:return: n with the maximum n/totient(n) ratio in the form: (n, totient(n), n/totient(n))
"""
max_ratio = (0, 0, 0)
for n in range(2, max_n+1, 2): # Check only even n
f = common.totient(n)
ratio = n/f
if max_ratio[-1] < ratio:
max_ratio = (n, f, ratio)
return max_ratio
def maximum_totient_ratio_odd(max_n):
"""Return number n < max_n with maximum n/totient(n) ratio.
Per analysis, the n with maximum n/totient(n) for any given range will be an even number that is twice an odd number
because of the property:
totient(2m) = 2*totient(m) if m is even
= totient(m) if m is odd
:param max_n: maximum n to check
:return: n with the maximum n/totient(n) ratio in the form: (n, totient(n), n/totient(n))
"""
max_ratio = (0, 0, 0)
for n in range(1, 1+max_n//2, 2): # Only odd n
f = common.totient(n)
ratio_2n = 2*n/f
if max_ratio[-1] < ratio_2n:
max_ratio = (2*n, f, ratio_2n)
return max_ratio
def solve():
return sum(totient(i) for i in xrange(2, 1000001))
def solve():
result = {n: float(n)/totient(n) for n in xrange(2, 1000001)}
return max(result.iteritems(), key=operator.itemgetter(1))[0]
def generate_totient_list(max_n):
"""
totient(m*n) = totient(m)*totient(n)*(d/totient(d)) where d = gcd(m, n)
totient(2*m) = 2*totient(m) if m is even
= totient(m) if m is odd
totient(n^m) == n^(m-1) * totient(n)
:param max_n:
:return:
"""
totient_list = [0] * (max_n+1)
prime_list = [2]
# Process 2 and multiples of 2 first
n = 2
totient_list[n] = n - 1
prev_phi = totient_list[n]
n2 = 2 * n
while n2 < max_n:
totient_list[n2] = 2 * prev_phi
prev_phi = totient_list[n2]
n2 *= 2
# Loop through rest of numbers
for n in range(3, max_n+1):
if totient_list[n] == 0:
if common.is_prime_mr(n): # all primes > 2 are odd
totient_list[n] = n - 1
n2 = 2 * n
if n2 < max_n:
totient_list[n2] = totient_list[n]
prev_phi = totient_list[n]
n2 *= 2
while n2 < max_n:
totient_list[n2] = 2 * prev_phi
prev_phi = totient_list[n2]
n2 *= 2
# for p in prime_list:
# index = p * n
# if index < max_n:
# totient_list[index] = totient_list[p] * totient_list[n]
# else:
# break
# prime_list.append(n)
m = 2
index_prev = n**(m - 1)
index = n**m
while index < max_n:
if totient_list[index] == 0:
totient_list[index] = index_prev * totient_list[n]
m += 1
index_prev = index
index = n**m
for m in range(3, n):
index = n * m
if index < max_n:
if totient_list[index] == 0:
totient_list[index] = totient_list[m] * totient_list[n]
else:
break
else: # n is not prime
totient_list[n] = common.totient(n)
prev_phi = totient_list[n]
n2 = 2 * n
if (n % 2 == 0) and (n2 < max_n): # if n is odd
if totient_list[n2] == 0:
totient_list[n2] = prev_phi
n2 *= 2
while n2 < max_n:
if totient_list[n2] == 0:
totient_list[n2] = 2 * prev_phi
prev_phi = totient_list[n2]
n2 *= 2
else:
prev_phi = totient_list[n]
n2 = 2 * n
if (n % 2 == 1) and (n2 < max_n): # if n is odd
if totient_list[n2] == 0:
totient_list[n2] = prev_phi
n2 *= 2
while n2 < max_n:
if totient_list[n2] == 0:
totient_list[n2] = 2 * prev_phi
prev_phi = totient_list[n2]
n2 *= 2
return totient_list
print('left_ordered_fraction((2, 5), 8) =', left_ordered_fraction((2, 5), 8))
print('left_ordered_fraction((3, 7), 8) =', left_ordered_fraction((3, 7), 8))
print('left_ordered_fraction((1, 2), 8) =', left_ordered_fraction((1, 2), 8))
print('left_ordered_fraction((3, 7), 10**6) =', left_ordered_fraction((3, 7), 10**6))
elif problem_num == 72:
print()
for z in range(2, 9):
print('count_reduced_fractions(', z, ') =', count_reduced_fractions(z))
# print('count_reduced_fractions(10**3) =', count_reduced_fractions(10**3))
print()
for z in range(2, 9):
print('list_reduced_fractions(', z, ') =', list_reduced_fractions(z))
print()
zlist = generate_totient_list(40)
for z in range(2, 41):
print(' ', z, zlist[z], common.totient(z))
print()
z = 10**3
print('sum(generate_totient_list(', z, ')) =', sum(generate_totient_list(z)))
elif problem_num == 73:
print()
elif problem_num == 74:
print()
elif problem_num == 75:
print()
elif problem_num == 76:
print()
elif problem_num == 77:
print()
elif problem_num == 78:
print()