def get_totient_maximum_below_bound(n):
table = [0 for i in range(n + 1)]
primes = [i for i in eu.sieve_of_eratosthenes(n) if i != 0]
p_se = set(primes)
for i in range(2, len(table)):
table[i] = i / phi(i, primes, p_se)
return table.index(max(table[8:]))
def get_sum_of_distinct_squarefree_nums_in_first_n_rows_of_pascal_triangle(n):
values = set(i for row in get_first_n_rows_of_pascal_triangle(n)
for i in row)
bound = math.ceil(math.sqrt(max(values)))
primes_squares = set(i**2 for i in eu.sieve_of_eratosthenes(bound)
if i != 0)
return sum(i for i in values if is_squarefree(i, primes_squares))
def get_three_primes_with_four_digit_chained_term():
primes = [get_key(i) for i in eu.sieve_of_eratosthenes(10000) if i > 1000]
s = set(primes)
dic = {p: 0 for p in s}
for p in primes:
dic[p] += 1
print(len(dic))
def x(bound):
prims = [i for i in eu.sieve_of_eratosthenes(bound) if i != 0]
i = 3 * 5 * 7 * 11 - 1
last = 0
while True:
last += gett(i, prims)
if last == 4:
exit(i - 3)
if last < 0:
last = 0
i += 1
def num_of_sum_of_primes_square_cube_and_fourth_power_below_bound(n):
primes = eu.sieve_of_eratosthenes(math.ceil(math.sqrt(n)))
return get_number_of_sums_below_bound(n, [i for i in primes if i != 0])
def get_sum_of_truncatable_primes_below_bound(n):
primes = {str(i) for i in eu.sieve_of_eratosthenes(n) if i != 0}
return sum(
int(p) for p in primes
if is_truncatable_prime(p, primes) and int(p) >= 23)
def get_consecutive_prime_sum_below_bound(n):
primes_in_order = [i for i in eu.sieve_of_eratosthenes(n) if i != 0]
return get_longest_consecutive_prime_sum(primes_in_order)
def get_smallest_odd_composite_that_is_not_the_sum_of_prime_and_twice_a_square(
bound):
primes = [i for i in eu.sieve_of_eratosthenes(bound) if i != 0]
double_squares = [2 * i**2 for i in range(math.ceil(math.sqrt(bound / 2)))]
odd_composites = {i + j for i in primes for j in double_squares}
return min(i for i in range(3, bound, 2) if i not in odd_composites)
def get_num_of_circular_primes_below_bound(n):
primes = {str(i) for i in eu.sieve_of_eratosthenes(n) if i != 0}
return sum(1 for i in primes if is_circular_prime(i, primes))
def get_sum_of_primes_below_bound(bound):
return sum(eu.sieve_of_eratosthenes(bound))
def get_sum_of_prime_generating_integers_below_n(n):
primes = set(eu.sieve_of_eratosthenes(n))
return sum(i for i in range(2, 100000001, 4)
if is_prime_generating(i, primes)) + 1
def get_max_n_digit_pandigital_prime():
return max(i for i in eu.sieve_of_eratosthenes(7654322)
if i != 0 and is_pandigital(i))