def sod(n):
"""Calculate sum of divisors of n"""
prod = 1
for k in primes.xprimes(int(n**0.5)+1):
p = 1
while n % k == 0:
p = p*k+1
n /= k
prod *= p
# n has a prime divisor > sqrt(n)
if n > 1:
prod *= 1+n
return prod;
def prime_factors(n, hint=None):
"""Return the prime factorization of in the form [(p,e), ...]"""
if not hint: hint = int(n ** 0.5) + 1
factors = []
for p in primes.xprimes(hint):
# Found all prime factors
if n <= 1: break
while 0 == n % p:
if not factors or factors[-1][0] != p:
factors.append([p,1])
else:
factors[-1][-1] += 1
n /= p
if n > 1:
#print "Hint was too small, %s likely prime" % n
if n < int(n ** 0.5) + 1:
factors += prime_factors(n, int(n ** 0.5) + 1)
else:
factors += prime_factors(n, n)
return factors
def gen_circular_primes(limit):
return [p for p in primes.xprimes(limit) if circular(p)]
def solve():
return sum(primes.xprimes(2000000))
def xfactor(num):
for prime in primes.xprimes(max=num):
while num % prime == 0:
yield prime
num /= prime