def combine_primes_below(r, N, primes):
pos = range(r)
L = len(primes)
solutions = []
if r > L:
return solutions
while True:
ps = [primes[j] for j in pos]
prod = product_of(ps)
if prod <= N:
solutions.append(ps)
index = -1
for i in range(0, r):
if pos[r-1-i] == L-1-i:
prod = product_of(ps[:r-1-i])
continue
elif prod > N:
prod = product_of(ps[:r-1-i])
continue
else:
index = r-1-i
break
if index == -1:
break
pos[index] += 1
for i in range(index+1, r):
pos[i] = pos[i-1] + 1
return solutions
def solve(number, size_of_subset):
largest_product = 0
for subset in consecutive_subsets_of(str(number), size_of_subset):
product = product_of([int(i) for i in subset])
if product > largest_product:
largest_product = product
return largest_product
def prime_powers_below(primes_base, N):
# TODO code has to be fixed, is just an estimate for ceiling N
prod = product_of(primes_base)
prime_powers_base = [powers_below(p, (N*p)/prod) for p in primes_base]
for factorization in product(prime_powers_base):
if product_of_factorization(factorization) <= N:
yield factorization
def solve(N):
primes = find_product_primes_below(N)
factors_set = [[n for n in powers_below(p, N, 1)] for p in primes]
max_frac = 0.0
max_number = 0
for factors in produce(factors_set):
n = product_of(factors)
if n > N:
continue
phi = product_of([num - num/p for (num,p) in zip(factors, primes)])
frac = float(n)/phi
if frac > max_frac:
print(n, frac, factors)
max_frac = frac
max_number = n
return max_number
def solve(ceiling):
factors = []
for p in primes_until(ceiling):
factor = p**max_power_of_x_below_y(p, ceiling)
factors.append(factor)
return product_of(factors)
def get_max_product_in_lines(lines, nums):
max_product = 0
for subset in consecutive_subsets_of(lines, nums):
p = product_of(subset)
max_product = max(p, max_product)
return max_product
"""
An irrational decimal fraction is created by concatenating the positive integers:
0.123456789101112131415161718192021...
It can be seen that the 12th digit of the fractional part is 1.
If d_(n) represents the n-th digit of the fractional part, find the value of the following expression:
d_(1) x d_(10) x d_(100) x d_(1000) x d_(10000) x d_(100000) x d_(1000000)
"""
from projecteuler import product_of
if __name__ == '__main__':
limit = 10**6
fraction = ""
for i in range(1, limit):
fraction += str(i)
if(len(fraction) > limit):
break
indexes = [10**i for i in range(7)]
result = product_of([int(fraction[i-1]) for i in indexes])
print("The value of the expression in problem 40 is %(result)d" % vars())
def product_of_factorization(pps):
return product_of([x**exp for (x,exp) in pps])