def solve():
dp = [0] * (N + 1)
dp[0] = 1
for i, p in enumerate(list_primes(N)):
for j in xrange(p, N + 1):
dp[j] += dp[j - p]
return [i for i, num_ways in enumerate(dp) if num_ways > 5000][0]
def solution(): ceiling = 100000 # Arbitrary initial cutoff primes = common.list_primes(ceiling) def find_set_sum(prefix, targetsize, sumlimit): if len(prefix) == targetsize: return sum(primes[i] for i in prefix) else: istart = 0 if (len(prefix) == 0) else (prefix[-1] + 1) for i in range(istart, len(primes)): if primes[i] > sumlimit: break if all((is_concat_prime(i, j) and is_concat_prime(j, i)) for j in prefix): prefix.append(i) result = find_set_sum(prefix, targetsize, sumlimit - primes[i]) prefix.pop() if result is not None: return result return None # Tests whether concat(primes[x], primes[y]) is a prime number, with memoization. @common.memoize def is_concat_prime(x, y): return is_prime(int(str(primes[x]) + str(primes[y]))) # Tests whether the given integer is prime. The implementation performs trial division, # first using the list of primes named 'primes', then switching to simple incrementation. # This requires the last number in 'primes' (if any) to be an odd number. def is_prime(x): if x < 0: raise ValueError() elif x in (0, 1): return False else: end = common.sqrt(x) for p in primes: if p > end: break if x % p == 0: return False for i in range(primes[-1] + 2, end + 1, 2): if x % i == 0: return False return True sumlimit = ceiling while True: setsum = find_set_sum([], 5, sumlimit - 1) if setsum is None: # No smaller sum found return str(sumlimit) sumlimit = setsum
def solve():
primes = list_primes(50000000)
result = set()
for p in primes:
for q in primes:
n = p * q
if n > 100000000:
break
else:
result.add(n)
return len(result)
def solve():
primes = list_primes(1000000)
result = {}
for x in primes[:300:-1]:
for y in primes[300:]:
n = x * y
if n > 10000000:
break
phi = totient(n)
if is_permutation(n, phi):
result[n] = float(n) / phi
return min(result.iteritems(), key=operator.itemgetter(1))[0]
def do_first_pass():
mr = MillerRabin()
concatenates = defaultdict(bool)
primes = list_primes(LIMIT)
pairs = set()
for i, p1 in enumerate(primes):
for p2 in primes[i + 1:]:
if mr.is_prime(int(str(p1) + str(p2))) and mr.is_prime(
int(str(p2) + str(p1))):
pairs.add(frozenset([p1, p2]))
concatenates[(p1, p2)] = True
concatenates[(p2, p1)] = True
return pairs, reduce(frozenset.union, pairs), concatenates
def solution():
ceiling = 50000000
primes = common.list_primes(common.sqrt(ceiling))
sums = {0}
for i in range(2, 5):
newsums = set()
for p in primes:
q = p**i
if q > ceiling:
break
for x in sums:
if x + q <= ceiling:
newsums.add(x + q)
sums = newsums
return str(len(sums))