def divisors_arithmetic_sequence_bf(limit_n, solutions):
total = 0
primes_index = prime_factors(3 * limit_n, False)
print('Primes calculated!')
primes = prime_factors(10**4)
t0 = time()
p = 1
for n in range(1, limit_n + 1):
if n > p * 10**6:
print('Now in: ', p)
p += 1
subtotal = 0
k = 3 * n
if primes_index[k] == 0:
if n % 4 in (0, 3):
sqrt_k = math.sqrt(k)
factors = list(decompose_primes(n, primes))
print(k, n % 4, 'Candidate', factors)
factors = decompose_primes(n, primes)
for d in range(1, math.ceil(sqrt_k)):
if k % d == 0:
if (d % 2) + ((k // d) % 2) != 1:
if check_compatibility(d, k // d, k):
subtotal += 1
if subtotal > solutions:
break
if subtotal == solutions:
print(k, list(factors))
total += 1
t1 = time()
print('Total time to get solution: ', t1 - t0)
return total
def solve():
n = 647
while(n):
if(len(prime_factors(n)) == 4 and
len(prime_factors(n+1)) == 4 and
len(prime_factors(n+2)) == 4 and
len(prime_factors(n+3)) == 4):
return n
n += 4
def combinations_over_value(limit_value):
primes = prime_factors(100)
primes_index = prime_factors(100, False)
value = 3
index = 0
solution = find_combs(primes, value, index)
while solution <= limit_value:
if primes_index[value] == 1:
index += 1
value += 1
solution = find_combs(primes, value, index)
return value
def num_lattice_points_fast(n):
# count with inclusion-exclusion
if n == 1:
return 1
if (n + 3) % 2 == 1:
return 0
a = (n + 3) / 2
half_a = (a - 1) / 2
first_lattice_point_x = 3 - (a % 3)
prime_factors_a = [p for p, m in utils.prime_factors(a)]
# A is every 3rd point on the line in the lattice, B1 is all points w/ x
# coord divisible by prime factor i
# #(A and not B) = #A - #(A and (B1 or B2 or ...))
# = #A - #((A and B1) or (A and B2) or ..)
# = #A - (#(A and B1) + #(A and B2) + ... - #(A and B1 and B2) - ...)
num_lattice_points_half = 0
factor_sets = utils.power_set(prime_factors_a)
for factors in factor_sets:
sign = 1 if (len(factors) % 2 == 0) else -1
pr = [(3, first_lattice_point_x)] + [(f, 0) for f in factors]
num_lattice_points_half += sign * count_elts_in_range_with_remainders(first_lattice_point_x, half_a + 1, pr)
return num_lattice_points_half * 2
def number_of_consec_primes(a, b):
n = 0
while True:
formula = n * n + a * n + b
if sum(prime_factors(formula)) != formula:
return n
n += 1
def prime_consecutive_sum(limit_value):
primes = prime_factors(limit_value, True)
consecutives = -float('inf')
max_prime = 0
for index in range(len(primes)-1, -1,-1):
num = primes[index]
if num < 990000:
return max_prime
starting_index = 0
while True:
current_index = starting_index
if index-starting_index+1<consecutives or starting_index==index:
break
total = 0
while True:
total += primes[current_index]
if total>num or current_index==index:
break
if total==num:
if consecutives < current_index-starting_index+1:
consecutives = current_index-starting_index+1
max_prime = num
print(num,consecutives)
break
current_index += 1
starting_index += 1
def pb3():
"""
Problem 3 : Largest prime factor
See utils.prime_factors for having the list of prime numbers.
Generates the list of prime factors and takes the last element.
"""
return utils.prime_factors(600851475143)[-1]
def product_pairs(n):
'''
2. This function solves the problem finding every possible pair of factors given its prime decomposition
'''
primes = prime_factors(1000)
prime_factors = decompose_primes(n, primes, as_dict=False)
prime_factors = [1] + prime_factors
pairs = {}
for r1 in range(1, len(prime_factors) // 2 + 1):
for comb1 in combinations([i for i in range(len(prime_factors))], r1):
a = 1
for idx in comb1:
a *= prime_factors[idx]
set_a = set(decompose_primes(a, primes))
complement = [
i for i in range(len(prime_factors)) if i not in comb1
]
for r2 in range(r1, len(complement) + 1):
for comb2 in combinations(complement, r2):
b = 1
for idx in comb2:
b *= prime_factors[idx]
if set_a.isdisjoint(set(decompose_primes(b, primes))):
if (a, b) not in pairs and (b, a) not in pairs:
pairs[(a, b)] = 1
return len(pairs) + 1
def get_n_with_maximum_ratio(limit_value):
primes = prime_factors(100)
div = 1
for prime in primes:
div *= prime
if div > limit_value:
return div // prime
def arithmetic_sequence():
primes = prime_factors(10**6, False)
perm_used = [0] * 10000
for value in range(1001, 9987):
perm4 = elements_perm_k(str(value), 4)
if perm_used[int(perm4[0])] == 0:
for a in range(0, len(perm4) - 2):
if primes[int(perm4[a])] and int(perm4[a]) > 1000:
for b in range(a + 1, len(perm4) - 1):
if primes[int(perm4[b])] and int(perm4[b]) > 1000:
d2 = int(perm4[b]) - int(perm4[a])
for c in range(b + 1, len(perm4)):
if primes[int(
perm4[c])] and int(perm4[c]) > 1000:
d1 = int(perm4[c]) - int(perm4[b])
if d1 == d2:
if (perm4[a], perm4[b], perm4[c]) != (
'1487', '4817', '8147'):
return perm4[a] + perm4[b] + perm4[
c]
else:
continue
else:
continue
else:
continue
else:
continue
for value in perm4:
perm_used[int(value)] = 1
def num_lattice_points_slow(n):
if n == 1:
return 1
if (n + 3) % 2 == 1:
return 0
a = (n + 3) / 2
half_a = (a - 1) / 2
first_lattice_point_x = 3 - (a % 3)
prime_factors_a = [p for p, m in utils.prime_factors(a)]
print prime_factors_a
# count by looping
num_lattice_points_half = 0
for x in xrange(first_lattice_point_x, half_a + 1, 3):
x_y_relatively_prime = True
for p in prime_factors_a:
if (x % p == 0):
x_y_relatively_prime = False
break
if x_y_relatively_prime:
num_lattice_points_half += 1
result = 2 * num_lattice_points_half
return result
def pythagorean_triplets(limit_n):
primes = prime_factors(1000)
total = 0
t0 = time()
for m in range(2, limit_n + 1):
if 2 * m**2 + 2 * m >= limit_n:
break
factors = decompose_primes(m, primes, True)
inc = 1
if m % 2 == 0:
inc = 2
complete = False
for n in range(1, m, inc):
if inc == 1 and n % 2 == 1:
continue
are_coprime = True
for p in factors:
if n % p == 0:
are_coprime = False
break
if are_coprime:
a = m**2 - n**2
b = 2 * m * n
c = m**2 + n**2
if (a + b + c) >= limit_n:
break
if c % (b - a) == 0:
print(a, b, c)
total += limit_n // (a + b + c)
if complete:
break
t1 = time()
print('Time to reach solution: {0} sec'.format(t1 - t0))
return total
def problem_five():
c = Counter()
sum = 1
for i in xrange(21):
i_c = Counter(prime_factors(i))
c = c | i_c
for k, v in dict(c).iteritems():
sum *= int(pow(k, v))
return sum
def repunit_divisibility_improved(limit_n):
primes = prime_factors(10**6)
factors = []
for i in primes[3:]:
factor = gcd(i - 1, primes)
if (10**factor) % i == 1:
factors.append(i)
continue
if len(factors) == 40:
return sum(factors)
def prime_remainders(limit_res):
'''
returns the least value of n for which the first remainder first exceeds 10**10
'''
primes = prime_factors(10**6)
n = 1
for p in primes:
if 2 * n * p > limit_res and n % 2 != 0:
return n
n += 1
def ratio_tracker():
primes_index = prime_factors(10**6, False)
primes = prime_factors(10**6, True)
print('Prime factors stored!')
l = 7
terms = 13
prime_counter = 8
corner_values = [31, 37, 43]
while ((prime_counter / terms) > 0.1):
l += 2
terms += 4
for index in range(len(corner_values)):
corner_values[index] += (index + 1) * 2 + 4 * (l - 3)
if corner_values[index] < 10**6:
if primes_index[corner_values[index]]:
prime_counter += 1
else:
prime_counter += is_prime(corner_values[index], primes)
return l
def get_total_set(limit_value):
primes = prime_factors(limit_value)
primes_index = prime_factors(limit_value, False)
counter = 0
for d in range(2,limit_value+1):
if primes_index[d]==1:
counter += d-1
else:
tmp = d-1
factors = find_prime_factors(d,primes)
for e in range(0,len(factors)):
variations = combinations(factors, r=e+1)
for var in variations:
num = 1
for elem in var:
num *= elem
if (e+1)%2==0:
tmp += (d-1)//num
else:
tmp -= (d-1)//num
counter += tmp
return counter
def sum_product_set_length(limit_value):
sol = {}
k = [i for i in range(2, limit_value + 1)]
k_index = {}
for i in k:
k_index[i] = 1
primes = prime_factors(10000)
primes_index = prime_factors(100000, False)
decomposition = {}
i = 4
while k:
if primes_index[i] == 0:
tmp = decompose_and_group(i, primes)
for g in tmp:
tmp_k = (i - g[1]) + len(g[0])
if tmp_k not in sol:
if tmp_k in k_index:
sol[tmp_k] = i
k.remove(tmp_k)
i += 1
print(i)
return sum(set(sol.values()))
def find_counter_between(limit_value):
primes = prime_factors(limit_value)
primes_index = prime_factors(limit_value, False)
lower = 1 / 3
upper = 1 / 2
counter = 0
for i in range(2, limit_value + 1):
if primes_index[i] == 1:
counter += math.ceil(i / 2) - math.floor(i / 3) - 1
else:
sequence = 0
factors = find_prime_factors(i, primes)
for s in range(math.ceil(i / 3), math.floor(i / 2) + 1):
ins = True
for p in factors:
if s % p == 0:
ins = False
break
if ins:
sequence += 1
counter += sequence
return counter
def radicals(limit_n):
'''
finds radicals for every prime number below limit_n
'''
radicals = {}
primes = prime_factors(limit_n)
for i in range(1, limit_n):
radicals[i] = list(decompose_primes(i, primes, True).keys())
square_free = [0] * (limit_n + 1)
for p in primes:
for i in range(p * p, limit_n, p * p):
square_free[i] = 1
return radicals, square_free
def primes_square_cube(limit_n):
primes_index = prime_factors(limit_n + 1, False)
i = 0
n = 2
counter = 0
while True:
i = i + n
tmp = i * 3 + 1
if tmp > limit_n:
break
if primes_index[tmp] == 1:
counter += 1
n += 2
return counter
def solution_given_primes():
'''
3. This function solves the problem using the prime decomposition to calculate the number of valid solutions
'''
n = 180100
primes = prime_factors(200000)
while True:
dec_primes = decompose_primes(n**2, primes, as_dict=True)
divs = 1
for p in dec_primes:
divs *= (dec_primes[p] + 1)
if divs > 1999:
return n
break
n += 1
def get_n_permutation(limit_value):
primes = prime_factors(limit_value)
min_ratio = float('inf')
candidate = 0
for i in range(len(primes) - 1, 100, -1):
for j in range(i, 100, -1):
number = primes[i] * primes[j]
if number < 10**7:
phi_acum = number - 1 - sum([
1 for c in range(primes[i], number, primes[i])
]) - sum([1 for c in range(primes[j], number, primes[j])])
if sorted(str(number)) == sorted(str(phi_acum)):
if min_ratio > (number / phi_acum):
min_ratio = (number / phi_acum)
candidate = number
return candidate
def divisors_arithmetic_sequence(limit_n):
total = 0
factors = [1] + prime_factors(limit_n)[1:]
print('Primes calculated!')
t0 = time()
for p in factors:
if p % 4 == 3:
total += 1
if 4 * p < limit_n:
total += 1
else:
continue
if 16 * p < limit_n:
total += 1
t1 = time()
print('Total time to get solution: ', t1 - t0)
return total
def phytagorean_triangle(limit_value):
primes = prime_factors(int(math.sqrt(limit_value)) + 1)
visited = [0] * (limit_value + 1)
n = 1
while n < limit_value:
lower = n + 1
ns = n**2
for m in range(lower, limit_value, 2):
if mcd(n, m, primes):
total = sum([m**2 - ns, 2 * m * n, m**2 + ns])
if total <= limit_value:
for i in range(total, limit_value + 1, total):
visited[i] += 1
else:
break
n += 1
return sum([1 for i in visited if i == 1])
def prime_triplets_bf(limit_value):
primes = prime_factors(int(math.sqrt(limit_value)) + 1)
total = 0
values = []
for a in range(len(primes)):
for b in range(len(primes)):
pt1 = primes[a]**2 + primes[b]**3
if pt1 >= limit_value:
break
for c in range(len(primes)):
pt2 = pt1 + primes[c]**4
if pt2 >= limit_value:
break
total += 1
values.append(pt2)
return len(set(values))
def smallest_in_amicable_chain(limit_value):
primes = prime_factors(limit_value, False)
print('primes_calculated!')
sieve = sum_proper_divisors(limit_value, primes)
print('Sieve calculated')
max_len = 0
max_chain = []
visited = {}
for i in range(2, limit_value + 1):
if primes[i] == 0:
ch = amicable_chain(i, limit_value, sieve, primes)
if len(ch) > max_len:
max_len = len(ch)
max_chain = ch
if i % 10**5 == 0:
print(i, max_len, min(max_chain))
return min(max_chain)
def get_n(limit_value):
limit_sqrt = int(math.sqrt(limit_value))
primes = prime_factors(limit_sqrt)
ref = 3/7
diff = float('inf')
num = 0
for i in range(limit_value,0,-1):
pivot = int(3*(i/7))
for j in range(pivot+2,pivot-2,-1):
tmp = j/i
if tmp < ref:
if ref-tmp<diff:
if set(find_prime_factors(j,primes))!=set(find_prime_factors(i,primes)):
diff = ref-tmp
num = j
break
return num
def euclid_pythagorean_triple():
return_type = namedtuple('Data', 'abc a b c')
primes = prime_factors(500)
all_combinations = set([1]) # Not returned as a prime
for i in range(1, len(primes) + 1):
all_combinations.update(
[product(tuple_) for tuple_ in combinations(primes, i)])
valid_set = sorted(all_combinations)
for k in valid_set:
for n, i in enumerate(valid_set):
for m in valid_set[i:]:
if gcd(m, n) == 1 and k * m * (m + n) == 500:
return return_type(k**3 * 2 * m * (m**4 - n**4),
k * (m**2 - n**2), 2 * m * n * k,
k * (m**2 + n**2))
def primes_runs():
primes = prime_factors(100005)
m = {}
n = {}
s = {}
digits= '0123456789'
for d in digits:
m[d]=0
n[d]=0
s[d]=0
digits_number = 10
visited = {}
for d1 in digits:
ar = d1*(digits_number-1)
for d2 in ''.join([d for d in digits if d!=d1]):
for idx in range(len(n)+1):
num = ar[:idx]+d2+ar[idx:]
if num[0]=='0':
continue
num_int =int(num)
if num not in visited:
visited[num]=1
if is_prime(num_int,primes):
max_checker(num,m,n,s)
complement = ''.join([d for d in digits if m[d]!=digits_number-1])
for d1 in complement:
ar = d1*(digits_number-2)
for d2 in ''.join([d for d in digits if d!=d1]):
for idx in range(len(ar)+1):
num = ar[:idx]+d2+ar[idx:]
for d3 in ''.join([d for d in digits if d!=d1]):
for idx1 in range(len(num)+1):
num1 = num[:idx1]+d3+num[idx1:]
if num1[0]=='0':
continue
num1_int = int(num1)
if num1 not in visited:
visited[num1]=1
if is_prime(num1_int,primes):
max_checker(num1,m,n,s)
return sum(s.values())
def prime_triplets(limit_value):
primes = prime_factors(int(math.sqrt(limit_value)) + 1)
total = 0
values = []
for pivot in range(len(primes)):
for index in range(pivot):
if (primes[pivot]**2 + primes[index]**3 +
primes[index]**4) >= limit_value:
break
triplet_variation = triplet_generation(primes[index],
primes[pivot], limit_value)
total += len(triplet_variation)
values += triplet_variation
pivot_triplet = primes[pivot]**2 + primes[pivot]**3 + primes[pivot]**4
if pivot_triplet < limit_value:
total += 1
values.append(pivot_triplet)
return len(set(values))
def numbers_with_n_distinct_factors(num_distinct = 4):
'''
Find the first four integers each with four distinct prime factors
problem 47
'''
Counter = collections.Counter
number_factors = [[],[]]
for i in range(2,1000000):
first = [i**v for i,v in Counter(prime_factors(i)).items()]
number_factors.append(first)
if len(first) >= num_distinct:
test_set = set(first)
b = number_factors[-num_distinct:-1]
expected = num_distinct * 2
for c in b[::-1]:
test_set.update(c)
if len(test_set) < expected:
break
expected += num_distinct
else:
return number_factors[-num_distinct:]
def pe3(n=600851475143): '''What is the largest prime factor of the number 600851475143?''' return max(utils.prime_factors(n))
def is_primitive_root(k, n):
factors = prime_factors(n-1)
return all(pow(k, (n-1)//f[0], n) > 1 for f in factors)
def solve():
'''Main function'''
num = 600851475143
return max(prime_factors(num))
""" Project Euler Problem #3 ========================= The prime factors of 13195 are 5, 7, 13 and 29. What is the largest prime factor of the number 600851475143? """ from utils import prime_factors target = 600851475143 print max(prime_factors(target))
"""
Project Euler Problem 5
=======================
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 number that is evenly divisible by all of the numbers
from 1 to 20?
"""
from collections import Counter
from utils import prime_factors
UPPER = 20
all_factors = Counter()
for i in range(1,UPPER+1):
for prime, count in Counter(prime_factors(i)).items():
if count>all_factors[prime]:
all_factors[prime] = count
product = 1
for prime, count in all_factors.items():
product = product*(prime**count)
print product
def hamming(n,limit=None):
phi = utils.euler_totient(n)
pf = utils.prime_factors(phi, limit)
if max(pf) <= 5:
return True
return False
# # The prime factors of 13195 are 5, 7, 13 and 29. # # What is the largest prime factor of the number 600851475143 ? # # from utils import prime_factors VALUE = 600851475143 factors = prime_factors(VALUE) print "Largest Prime factor is: %s" % factors[-1]
def prob3():
return utils.prime_factors(600851475143)[-1::]