def main(top=1000000):
prime_found = None
# Generate a list with cumalitive sum
cum_sum_primes = [0]
cum_sum = 0
for p in primes_erat():
cum_sum += p
cum_sum_primes.append(cum_sum)
if cum_sum > top:
break
num_primes = len(cum_sum_primes)
num_terms = num_primes
while True:
for offset in range(0, 10):
if offset + num_terms >= num_primes:
break
sum_of_primes = cum_sum_primes[offset +
num_terms] - cum_sum_primes[offset]
if sum_of_primes < top and is_prime(sum_of_primes):
prime_found = sum_of_primes
break
if prime_found:
break
num_terms -= 1
return prime_found
def main(smaller_than=None):
if not smaller_than:
smaller_than = Fraction(15499, 94744)
# We know that phi(n)/n is the smallest when n is a primorial,
# so first look for Phi(priomorial)/primorial which is smaller than our
# target faction.
primorial = 1
prime_list = []
for prime in primes_erat():
primorial *= prime
prime_list.append(prime)
phi_primorial = reduce(operator.mul, ((p - 1) for p in prime_list))
ratio = Fraction(phi_primorial, primorial)
print(primorial, ratio)
if ratio < smaller_than:
break
#return primorial
# Then search for Phi(n)/n-1 which is smaller than our target fraction.
found = False
for n in range(primorial, primorial + 100000):
ratio = Fraction(phi(n), n - 1)
if n % 1000 == 0:
print(n, ratio)
if ratio < smaller_than:
found = True
break
if found:
return n
else:
return 0
def main(max_prime=1000000):
total = 0
count = 0
prime_list = set()
for n in primes_erat():
prime_list.add(n)
if n < 10:
continue
n = str(n)
is_truncatable = True
# Remove digits from the right
for l in range(len(n) - 1, 0, -1):
x = n[:l]
if not int(x) in prime_list:
is_truncatable = False
break
if not is_truncatable:
continue
# Remove digits from the left
for l in range(1, len(n)):
x = n[l:]
if not int(x) in prime_list:
is_truncatable = False
break
if not is_truncatable:
continue
# If all above test have passed, it's one of the numbers we're after
total += int(n)
count += 1
if count == 11:
break
return total
def main(limit=10**7):
i = 1
n = 1
min_ratio = None
min_n = None
for p1 in primes_erat():
for p2 in itertools.takewhile(lambda p: p < p1, primes_erat()):
n = p1 * p2
if n > limit:
break
phi_n = (p1 - 1) * (p2 - 1)
ratio = n / phi_n
if (min_ratio is None or ratio < min_ratio) and is_permutation(
n, phi_n):
min_ratio = ratio
min_n = n
if p1 * 2 > limit:
break
return min_n
def main():
pandigital = list(range(1, 10))
largest = None
for prime in primes_erat():
if prime < 10**6:
# numbers below 7 digits are not interesting
continue
elif prime > 10**7:
# pandigital numbers with 8 or 9 digits,
# are all divisible by 9 (non prime)
break
digits = str(prime)
if sorted(int(d) for d in digits) == pandigital[:len(digits)]:
largest = prime
return largest
def main():
results = {}
for prime in itertools.takewhile(lambda p: p < 9999, primes_erat()):
if prime >= 1000:
number_hash = ''.join(sorted(x for x in str(prime)))
if not results.get(number_hash, False):
results[number_hash] = []
results[number_hash].append(prime)
for number_hash, prime_list in results.items():
if len(prime_list) < 3:
continue
for a, b, c in itertools.combinations(prime_list, 3):
if (a, b, c) == (1487, 4817, 8147):
continue
if b - a == c - b:
return int(''.join(str(i) for i in (a, b, c)))
def main(max_prime=1000000):
prime_list = set(
p for p in itertools.takewhile(lambda x: x < max_prime, primes_erat()))
count = 0
for n in prime_list:
is_circular = True
for x in rotations(str(n)):
x = int(x)
if x == n:
continue
if x < max_prime and not x in prime_list:
is_circular = False
break
elif x >= max_prime and not is_prime(x):
is_circular = False
break
if is_circular:
count += 1
return count
def main():
return sum(itertools.takewhile(lambda x: x < 2000000, primes_erat()))
def main():
def tup_max(a, b):
if a[1] > b[1]:
return a
else:
return b
return reduce(tup_max, ((p, period_len_fraction(1, p)) for p in itertools.takewhile(lambda p: p < 1000, primes_erat())))[0]
def main():
return next(
itertools.islice(
itertools.takewhile(lambda p: p < 200000, primes_erat()), 10000,
None))