def odd_divisible_pandigitals():
small_primes = prime_sieve(20)
def sub_divides(number):
if all( sub % prime == 0 for sub, prime
in izip(substring_numbers(number,3,1),cycle(small_primes)) ):
return True
return False
for pandigital in ifilter(sub_divides,pandigitals(10)):
yield pandigital
def smallest_odd_composite():
def odd_numbers(start=1):
num = start
while True:
yield num
num += 2
primes = set(prime_sieve(10000))
odd_composites = ( i for i in odd_numbers(9)
if i not in primes and i < 10000)
for n in odd_composites:
if any( (n - (2 * base ** 2)) in primes
for base in xrange(1, (int(sqrt((n-2)/2))+1)) ):
continue
return n
def largest_prime_seq(limit):
primes = prime_sieve(limit)
prime_set = set(primes)
# determine the maximum allowable sequence length
test_num = 0
for max_seq_len, prime in enumerate(primes):
test_num += prime
if test_num > limit: break
for length in xrange(max_seq_len,0,-1):
for start in xrange(len(primes)-length):
seq = primes[start:start+length]
seq_sum = sum(seq)
if seq_sum >= limit: break
if seq_sum in prime_set:
return seq
# sum for a set of four primes with this property.
#
# Find the lowest sum for a set of five primes for which any two
# primes concatenate to produce another prime.
from itertools import dropwhile
from collections import defaultdict
from math import floor, log10
from p010 import prime_sieve
from p049 import is_prime
import psyco
psyco.full()
primes = prime_sieve(100000)
primes.remove(2) # Can never be in a set of more than one
primes.remove(5) # Can never be in a set of more than one
whitelist = defaultdict(set)
blacklist = defaultdict(set)
good_sets = defaultdict(list)
def is_concatenable_pair(x, y):
# if y in blacklist[x]:
# return False
if y in whitelist[x]:
return True
if not (is_concatenable_combo(x, y) and is_concatenable_combo(y, x)):
blacklist[x].add(y)
return False