def totient(n, factors=None):
if factors is None:
factors = prime_factorization(n).keys()
for fac in factors:
n //= fac
n *= fac - 1
return n
def totient(n, factors=None):
if factors is None:
factors = prime_factorization(n).keys()
for fac in factors:
n //= fac
n *= fac - 1
return n
def main():
square_free = {1}
current = [1]
for i in range(50):
current = next_pascal(current)
# print(current)
for j in current:
if all(exp < 2 for exp in prime_factorization(j).values()):
square_free.add(j)
print(sum(square_free))
def main():
number = 100
while True:
factorization = prime_factorization(number * (number - 1) // 2)
divisors = 1
for i in factorization.values():
divisors *= i + 1
if divisors > 500:
print(number * (number - 1) // 2)
break
number += 1
def decompose(n):
"""Decompose inverse square as sum of prime inverse squares"""
p_fact = sorted(prime_factorization(n).items())
facts_exp = [f**(e * 2) for f, e in p_fact]
coeffs = [1] * len(facts_exp)
prod = facts_exp[0]
for i in range(1, len(facts_exp)):
coeff2, coeff1 = extended_gcd(prod, facts_exp[i])
for j in range(i):
coeffs[j] *= coeff1
coeffs[i] = coeff2
prod *= facts_exp[i]
return list(zip(coeffs, facts_exp, [f for f, e in p_fact]))
def main():
coprime_pairs = farey(100)
coprime_pairs.sort()
for cand in coprime_pairs:
print(cand, sorted(prime_factorization(cand[0]).items()))
def reciprocal_solution(n):
# return sum(n*i % (i-n) == 0 for i in range(n+1, 2*n+1)) # slow
return (product(2 * i + 1 for i in prime_factorization(n).values()) + 1) // 2
def main():
coprime_pairs = farey(100)
coprime_pairs.sort()
for cand in coprime_pairs:
print(cand, sorted(prime_factorization(cand[0]).items()))
def reciprocal_solution(n):
# return sum(n*i % (i-n) == 0 for i in range(n+1, 2*n+1)) # slow
return (product(2 * i + 1
for i in prime_factorization(n).values()) + 1) // 2
from mathutil import prime_factorization
def print_factorization(factorization):
factors = sorted(factorization.keys())
print(" * ".join("{}^{}".format(fac, factorization[fac]) for fac in factors))
def main():
number = 100
while True:
factorization = prime_factorization(number * (number - 1) // 2)
divisors = 1
for i in factorization.values():
divisors *= i + 1
if divisors > 500:
print(number * (number - 1) // 2)
break
number += 1
if __name__ == '__main__':
from time import time
starting_time = time()
print_factorization(prime_factorization(1234567890))
main()
print("Time elapsed:", time() - starting_time, "seconds")
