def triangular_divisors(n):
if n%2==0:
af = factor(n/2)
bf = factor(n+1)
else:
af = factor((n+1)/2)
bf = factor(n)
triangular_factors = af + bf
ndivizors = 1
for p, e in triangular_factors:
ndivizors *= (e+1)
return ndivizors
def divisor_iter(n):
factorization = factor(n)
# factorization of the current divisor
divisor_fact = [(p, 0) for p, e in factorization]
def _next_divisor_fact():
""" count up from 2**0 * 3*0 * ... to the factorization"""
for i in xrange(len(factorization)):
dp, de = divisor_fact[i]
if de < factorization[i][1]:
divisor_fact[i] = (dp, de + 1)
return divisor_fact
else: #overflow
divisor_fact[i] = (dp, 0)
if i == len(factorization) - 1:
return None
def _compute_divisor():
mul = 1
for p, e in divisor_fact:
mul *= p**e
return mul
while True:
yield _compute_divisor()
if not _next_divisor_fact():
return
def pb12():
"""
What is the value of the first triangle number to have over five hundred divisors?
"""
for i in xrange(2, 10**9):
td = triangular_divisors(i)
if (td >= 500):
print i, td, i*(i+1)/2, factor(i*(i+1)/2)
break
def totient(n, cached_primes=None):
t = 1
for p, e in factor(n, cached_primes):
t *= p**(e - 1) * (p - 1)
return t