def cycle(i):
factors = prime_factors(i)
# Filter out all 2s and 5s
i = 1
for p in factors:
if p != 2 and p != 5:
i = i * p
if i == 1:
return 0
digits = 1
n = 9
while n % i != 0:
n = n * 10 + 9
digits = digits + 1
return digits
#!/usr/bin/env python3
'''
Largest prime factor
The prime factors of 13195 are 5, 7, 13 and 29.
What is the largest prime factor of the number 600851475143 ?
'''
from sys import argv
from prime import prime_factors
if __name__ == '__main__':
try:
target = int(argv[1])
except IndexError:
target = 600851475143
print(max(prime_factors(target)))
2520 is the smallest number that can be divided by each of the numbers from 1
to 10 without any remainder.
What is the smallest positive number that is evenly divisible by all of the
numbers from 1 to 20?
"""
from collections import defaultdict
from sys import argv
from prime import prime_factors
if __name__ == "__main__":
try:
target = int(argv[1])
except IndexError:
target = 20
powers = defaultdict(int)
for n in range(2, target + 1):
prime_powers = prime_factors(n)
for k in prime_powers:
powers[k] = max(powers[k], prime_powers[k])
total = 1
for k in powers:
total *= k ** powers[k]
print(total)
def num_divisors(x):
# Based on http://mathschallenge.net/index.php?section=faq&ref=number/number_of_divisors
factors = prime_factors(x)
return reduce(lambda x, y: x * (y + 1), factors.values(), 1)
import prime
def com(n,k):
return factorial(n)/(factorial(n-k)*factorial(k))
def com(n,k):
return factorial(n)/(factorial(n-k)*factorial(k))
b = (0,0)
e = (2,2)
p = [(0,1),(0,1),(1,0),(1,0)]
r = set([pp for pp in permutations(p)])
#print(r)
print(len(r))
# Too big
#p = [(0,1),(0,1),(0,1),(0,1),(0,1),(0,1),(0,1),(0,1),(0,1),(0,1),(1,0),(1,0),(1,0),(1,0),(1,0),(1,0),(1,0),(1,0),(1,0),(1,0)]
#r = set([pp for pp in permutations(p)])
#print(r)
#print(len(r))
print( com(52,5) )
print( com(500,5) )
res = com(2,1)*com(10,1)*com(2,1)*com(10,1)*com(2,1)*com(9,1)*com(2,1)*com(9,1)*com(2,1)*com(8,1)*com(2,1)*com(8,1)*com(2,1)*com(7,1)*com(2,1)*com(7,1)*com(2,1)*com(6,1)*com(2,1)*com(6,1)*com(2,1)*com(5,1)*com(2,1)*com(5,1)*com(2,1)*com(4,1)*com(2,1)*com(4,1)*com(2,1)*com(3,1)*com(2,1)*com(3,1)*com(2,1)*com(2,1)*com(2,1)*com(2,1)
print( res )
print( prime.prime_factors(res / 137846528820) )
print( prime.prime_factors(137846528820) )
# unsolved
The first two consecutive numbers to have two distinct prime factors are:
14 = 2 × 7
15 = 3 × 5
The first three consecutive numbers to have three distinct prime factors are:
644 = 2² × 7 × 23
645 = 3 × 5 × 43
646 = 2 × 17 × 19.
Find the first four consecutive integers to have four distinct prime factors.
What is the first of these numbers?
'''
from itertools import count
from prime import prime_factors
if __name__ == '__main__':
sequence = 0
for number in count(1):
if len(prime_factors(number)) == 4:
sequence += 1
if sequence == 4:
print(number - 3)
break
else:
sequence = 0
from prime import get_primes, prime_factors
max_prime = 300000
primes = get_primes(max_prime)
i = 0
while i < max_prime-3:
if len(prime_factors(i+3, primes)) == 4:
if len(prime_factors(i+2, primes)) == 4:
if len(prime_factors(i+1, primes)) == 4:
if len(prime_factors(i, primes)) == 4:
print(i)
else:
i += 1
else:
i += 2
else:
i += 3
else:
i += 4
Each tuple is create by taking a digit from :param digits: and removing the first instance of that digit from both i and j. If this would cause either i or j to be 0, then that tuple is skipped.
"""
return filter(
lambda x: x[0] > 0 and x[1] > 0,
((int(str(i).replace(d, '', 1)), int(str(j).replace(d, '', 1)))
for d in digits))
fract = [1, 1]
for i in range(10, 100):
i_digits = set(str(i))
for j in range(i + 1, 100):
if i % 10 == 0 and j % 10 == 0:
continue
j_digits = set(str(j))
inter = i_digits.intersection(j_digits)
for n, d in generate_test_fractions(i, j, inter):
if isclose(i / j, n / d):
print("{}/{} == {}/{}".format(i, j, n, d))
fract[0] *= n
fract[1] *= d
fract = cancel_common_factors(prime_factors(fract[0]), prime_factors(fract[1]))
print("Denominator of products: {}".format(expand_factorization(fract[1])))
print("Execution time: {}".format(time.time() - start_time))
divisibility_target = 500
def merge_factors(factorization_1, factorization_2):
"""Requires parameters in format returned by prime.prime_factors."""
factors = set(factorization_1).union(factorization_2)
return {
k: factorization_1.get(k, 0) + factorization_2.get(k, 0)
for k in factors
}
# the first constiuent of the closed form of a triangle number: n(n+1)/2
n = 1
n_factors = dict()
next_factors = prime_factors(n + 1)
factor_count = 1
while factor_count <= divisibility_target:
n += 1
n_factors = next_factors
next_factors = prime_factors(n + 1)
merged = merge_factors(n_factors, next_factors)
# There will always be an extra factor of 2 (see closed form), which is divided out
merged[2] -= 1
# The actual number of factors (prime and otherwise) is given by this product
factor_count = reduce(mul, (v + 1 for v in merged.values()))
from prime import prime_factors
distinct_primes = [1, 1, 1, 1]
n = 647
still_checking = True
while still_checking:
distinct_primes[0] = distinct_primes[1]
distinct_primes[1] = distinct_primes[2]
distinct_primes[2] = distinct_primes[3]
s = set(prime_factors(n))
distinct_primes[3] = len(s)
if distinct_primes[0] > 3 and distinct_primes[1] > 3 and distinct_primes[2] > 3 and distinct_primes[3] > 3:
still_checking = False
print(n)
else:
n += 1
#/bin/python # http://projecteuler.net/problem=3 # The prime factors of 13195 are 5, 7, 13 and 29. What is the largest prime factor of the number 600851475143? import prime fact = prime.prime_factors(600851475143) print( fact ) # [71, 839, 1471, 6857] print( max(fact) ) # 6857
def main():
print(max(prime_factors(600851475143)))
import prime print(prime.prime_factors(600851475143))
2520 is the smallest number that can be divided by each of the numbers from 1
to 10 without any remainder.
What is the smallest positive number that is evenly divisible by all of the
numbers from 1 to 20?
'''
from collections import defaultdict
from sys import argv
from prime import prime_factors
if __name__ == '__main__':
try:
target = int(argv[1])
except IndexError:
target = 20
powers = defaultdict(int)
for n in range(2, target + 1):
prime_powers = prime_factors(n)
for k in prime_powers:
powers[k] = max(powers[k], prime_powers[k])
total = 1
for k in powers:
total *= k ** powers[k]
print(total)