def ans():
size = 5
skip_to = 8300
for p in Prime.gen_nums():
if p < skip_to:
continue
nums = [q for q in Prime.gen_nums(p) if compatible(q, p)]
for tuple_ in combinations(nums, size - 1):
if remarkable(tuple_):
return p + sum(tuple_)
def ans():
for prime in Prime.gen_nums():
if prime < 120000:
continue
# Replace all combinations of digits
list_ = list(str(prime))
length = len(list_)
for count in range(length):
combs = combinations(range(length), count)
for indices in combs:
# Count the number of primes by replacing
# the selected digits with some other digit
generated_primes = set()
for replacement in range(10):
copy = list_.copy()
for i in indices:
copy[i] = str(replacement)
number = int(''.join(copy))
if Prime.contains(number):
generated_primes.add(number)
if 7 < len(generated_primes):
return min(generated_primes)
def ans():
truncatable = set()
for p in Prime.gen_nums():
if is_truncatable(p):
truncatable.add(p)
if len(truncatable) == 11:
break
return sum(truncatable)
def ans():
n = 1
for p in Prime.gen_nums():
next_ = n * p
if 1000000 < next_:
return n
n = next_
return n
def ans():
groups = defaultdict(set)
for prime in Prime.gen_nums(10000):
groups[''.join(sorted(str(prime)))].add(prime)
for set_ in groups.values():
for comb in combinations(set_, 3):
seq = sorted(list(comb))
if seq[0] < 1000:
continue
if (seq[2] + seq[0] == 2 * seq[1] and seq[0] != 1487):
return ''.join(str(n) for n in seq)
def ans():
prime_list = list(Prime.gen_nums(4000))
longest = (0, 0)
for i in range(len(prime_list)):
sum_ = 0
for j in range(i, len(prime_list)):
sum_ += prime_list[j]
if 1000000 <= sum_:
break
if Prime.contains(sum_) and longest[1] < j - i + 1:
longest = (sum_, j - i + 1)
return longest[0]
def ans():
divisors = list(Prime.gen_nums(18))
sum_ = 0
for p in gen_pandigitals(from_=0, to=9):
has_property = True
for i in range(1, 8):
dividend = int(str(p)[i:i + 3])
if dividend % divisors[i - 1] != 0:
has_property = False
break
if has_property:
sum_ += p
return sum_
def ans():
circular_primes = set([2])
for p in Prime.gen_nums(1000000):
if any(x in str(p) for x in '02468'):
continue
is_circular = True
for i in range(len(str(p))):
if not Prime.contains(int(str(p)[i:] + str(p)[:i])):
is_circular = False
break
if is_circular:
circular_primes.add(p)
return len(circular_primes)
def ans():
# The insight is that n / phi(n) is minimized when n has a few, large prime
# factors (n can't be prime, since prime numbers don't satisfy the
# permutation criteria). Also, note that we don't actually have to compute
# all of the numbers coprime with n to determine phi(n); instead we can use
# the fact that n is a product of two primes to calculate phi(n) in
# constant time.
min_n = None
min_value = None
primes = Prime.gen_nums(10**4)
combos = combinations(primes, 2)
for one, two in list(combos):
n = one * two
if 10**7 <= n:
continue
num_coprimes = one * two - one - two + 1
if sorted(str(n)) == sorted(str(num_coprimes)):
value = n / num_coprimes
if not min_value or value < min_value:
min_n = n
min_value = value
return min_n
def ans():
return sum(Prime.gen_nums(2000000))