def main():
for i, factors in enumerate(prime_factorization_under(10 ** 6)):
number = i + 2
num_solution = (product(2 * i + 1 for i in factors.values()) + 1) // 2
# print(number, num_solution)
if num_solution > 1000:
print(number, num_solution)
break
def main():
for i, factors in enumerate(prime_factorization_under(10**6)):
number = i + 2
num_solution = (product(2 * i + 1 for i in factors.values()) + 1) // 2
# print(number, num_solution)
if num_solution > 1000:
print(number, num_solution)
break
def check_n(num, factors):
num_of_fac = len(factors)
if not primality[num + 1]:
return False
for i in range(1, num_of_fac//2+1):
for comb in combinations(range(num_of_fac), i):
part1 = product(factors[j] for j in comb)
if not primality[part1 + num//part1]:
return False
return True
def check_n(num, factors):
num_of_fac = len(factors)
if not primality[num + 1]:
return False
for i in range(1, num_of_fac // 2 + 1):
for comb in combinations(range(num_of_fac), i):
part1 = product(factors[j] for j in comb)
if not primality[part1 + num // part1]:
return False
return True
def main():
N = 4 * 10**6
N2 = N * 2 + 1
primes = prime_under(50)
# print(log(N2, 3), len(primes), list(enumerate(primes)), sep='\n')
choice = []
for i3 in range(ceil(log(N2, 3)) + 1):
current3 = 3**i3
if current3 > N2:
choice.append([3] * i3)
break
for i5 in range(ceil(log(N2, 5)) + 1):
current5 = current3 * 5**i5
if current5 > N2:
choice.append([3] * i3 + [5] * i5)
break
for i7 in range(ceil(log(N2, 7)) + 1):
current7 = current5 * 7**i7
if current7 > N2:
choice.append([3] * i3 + [5] * i5 + [7] * i7)
break
for i9 in range(ceil(log(N2, 9)) + 1):
current9 = current7 * 9**i9
if current9 > N2:
choice.append([3] * i3 + [5] * i5 + [7] * i7 +
[9] * i9)
break
for i11 in range(ceil(log(N2, 11)) + 1):
current11 = current9 * 11**i11
if current11 > N2:
choice.append([3] * i3 + [5] * i5 + [7] * i7 +
[9] * i9 + [11] * i11)
break
# print(len(choice))
# min_n = 10 ** 20 # inf
# min_comp = []
# for seq in choice:
# current_n = product(i**(j//2) for i, j in zip(primes, reversed(seq)))
# # print(current_n, seq)
# if current_n < min_n:
# min_n = current_n
# min_comp = seq
# print(min_comp)
min_n = min(
product(i**(j // 2) for i, j in zip(primes, reversed(seq)))
for seq in choice)
print(min_n)
def main():
N = 4 * 10 ** 6
N2 = N * 2 + 1
primes = prime_under(50)
# print(log(N2, 3), len(primes), list(enumerate(primes)), sep='\n')
choice = []
for i3 in range(ceil(log(N2, 3))+1):
current3 = 3 ** i3
if current3 > N2:
choice.append([3]*i3)
break
for i5 in range(ceil(log(N2, 5))+1):
current5 = current3 * 5 ** i5
if current5 > N2:
choice.append([3]*i3 + [5]*i5)
break
for i7 in range(ceil(log(N2, 7))+1):
current7 = current5 * 7 ** i7
if current7 > N2:
choice.append([3]*i3 + [5]*i5 + [7]*i7)
break
for i9 in range(ceil(log(N2, 9))+1):
current9 = current7 * 9 ** i9
if current9 > N2:
choice.append([3]*i3 + [5]*i5 + [7]*i7 + [9]*i9)
break
for i11 in range(ceil(log(N2, 11))+1):
current11 = current9 * 11 ** i11
if current11 > N2:
choice.append([3]*i3 + [5]*i5 + [7]*i7 + [9]*i9 + [11]*i11)
break
# print(len(choice))
# min_n = 10 ** 20 # inf
# min_comp = []
# for seq in choice:
# current_n = product(i**(j//2) for i, j in zip(primes, reversed(seq)))
# # print(current_n, seq)
# if current_n < min_n:
# min_n = current_n
# min_comp = seq
# print(min_comp)
min_n = min(product(i**(j//2) for i, j in zip(primes, reversed(seq)))
for seq in choice)
print(min_n)
def main():
# brute firce
# count = 0
# for i in range(1, SIZE):
# count += div7_row(i)
# print(count)
# print(SIZE * (SIZE + 1) // 2 - count)
# POW = ceil(log(SIZE, 7))
# pow7 = [7**i for i in range(1, POW)]
# triangle = [(i - 1) * i // 2 for i in pow7]
# whole_triangle = [triangle[0]]
# for i in range(1, POW - 1):
# whole_triangle.append(whole_triangle[-1] * 21 * 3 + triangle[i])
# print(pow7)
# print(triangle)
# print(whole_triangle)
# # top level
# covered = (7 ** (POW - 1) * 2)
# total = whole_triangle[-1] * (SIZE // covered)
# print(total)
# print(SIZE - covered)
# by Lucus's Thm
count = 0
row_base7 = [0]
for row in range(0, SIZE):
count += product(i + 1 for i in row_base7)
row_base7[0] += 1
for d in range(len(row_base7)):
if row_base7[d] == 7:
row_base7[d] = 0
if d < len(row_base7) - 1:
row_base7[d + 1] += 1
else:
row_base7.append(1)
break
else:
break
print(count)
def main():
# brute firce
# count = 0
# for i in range(1, SIZE):
# count += div7_row(i)
# print(count)
# print(SIZE * (SIZE + 1) // 2 - count)
# POW = ceil(log(SIZE, 7))
# pow7 = [7**i for i in range(1, POW)]
# triangle = [(i - 1) * i // 2 for i in pow7]
# whole_triangle = [triangle[0]]
# for i in range(1, POW - 1):
# whole_triangle.append(whole_triangle[-1] * 21 * 3 + triangle[i])
# print(pow7)
# print(triangle)
# print(whole_triangle)
# # top level
# covered = (7 ** (POW - 1) * 2)
# total = whole_triangle[-1] * (SIZE // covered)
# print(total)
# print(SIZE - covered)
# by Lucus's Thm
count = 0
row_base7 = [0]
for row in range(0, SIZE):
count += product(i + 1 for i in row_base7)
row_base7[0] += 1
for d in range(len(row_base7)):
if row_base7[d] == 7:
row_base7[d] = 0
if d < len(row_base7) - 1:
row_base7[d + 1] += 1
else:
row_base7.append(1)
break
else:
break
print(count)
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 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