def main(verbose=False):
primes = sieve(100)
MAX_n = 10 ** 16
product_factor_pairs = [(1, 0)]
product_hash = {0: [1]}
for num_factors in range(1, 13 + 1):
product_hash[num_factors] = []
# (1,0) becomes
# (1,0), (p, 1) becomes
# (1,0), (p, 1), (q,1), (q*p, 2) etc.
for prime in primes:
to_add = []
for product, num_factors in product_factor_pairs:
if prime * product < MAX_n:
to_add.append((prime * product, num_factors + 1))
product_hash[num_factors + 1].append(prime * product)
product_factor_pairs += to_add
result = 0
sign = -1
for num_factors in range(4, 13 + 1):
sign = -sign
PIE_factor = sign * choose(num_factors - 1, 3)
current_sum = 0
for product in product_hash[num_factors]:
current_sum += MAX_n / product # integer division
result += PIE_factor * current_sum
return result
def main(verbose=False):
expected_hash = {0: 0}
for n in range(1, 32 + 1):
to_add = 2 ** n
for k in range(1, n + 1):
to_add += choose(n, k) * expected_hash[n - k]
expected_hash[n] = (to_add * 1.0) / (2 ** n - 1)
return round(expected_hash[32], 10)
def main(verbose=False):
n = 100
return choose(n + 10, 10) + choose(n + 9, 9) - 10 * n - 2
def main(verbose=False):
# In an n x m grid there are (n + m) C m = (n + m) C n such paths.
return choose(20 + 20, 20)
