def p087():
LIM = 50000000
p4s = list(primes_less_than(int(math.pow(LIM, 1.0 / 4.0) + 1)))
p3s = list(primes_less_than(int(math.pow(LIM, 1.0 / 3.0) + 1)))
p2s = list(primes_less_than(int(math.pow(LIM, 1.0 / 2.0) + 1)))
results = set()
for p4 in p4s:
for p3 in p3s:
for p2 in p2s:
results.add(p4 ** 4 + p3 ** 3 + p2 ** 2)
# print (p2,p3,p4,p4**4+p3**3+p2**2)
return sum(1 for r in results if r < LIM)
def using_sieve():
ps = list(primes_less_than(LIMIT+1))
totients = [n for n in xrange(0, LIMIT+1)]
totients[0] = totients[1] = 0
for p in ps:
for n in xrange(p, LIMIT+1, p):
totients[n] = totients[n] * (p-1) / p
return sum(totients)
def p077():
denoms = list(reversed(list(primes_less_than(100))))
for i in xrange(10, 100):
count = combos(i, 0, denoms)
if i in denoms:
count -= 1
if count > 5000:
return i
def p046():
LIMIT = 100000
good = set(range(9,LIMIT,2))
squares = [2*n*n for n in xrange(1, int(math.sqrt(LIMIT//2))+1)]
for p in primes_less_than(LIMIT):
for s in squares:
good.discard(p)
good.discard(p+s)
return list(good)[0]
def fast(count, limit):
divider = reduce(operator.mul, list(primes_less_than(100))[:count-1], 1)
factors = dict()
for p in primes_less_than(limit//divider):
for n in xrange(p, limit, p):
factors[n] = factors.setdefault(n, 0) + 1
candidates = set(k for k,v in factors.iteritems() if v==count)
for c in sorted(list(candidates)):
found = True
for d in xrange(c+1, c+count):
if d not in candidates:
found = False
break
if found:
return c
return 0
def p049():
four_digit_primes = [p for p in primes_less_than(10000) if p > 999]
perms = dict()
for f in four_digit_primes:
perms.setdefault(''.join(sorted(str(f))), []).append(f)
candidates = [v for k, v in perms.iteritems() if len(v) > 2]
for c in candidates:
for ariths in find_ariths(c):
if ariths[0] != 1487:
return int("%d%d%d" % (ariths[0], ariths[1], ariths[2]))
def p050():
limit = 1000000
ps = list(primes_less_than(limit))
l_l = 0
l_p = 0
l = len(ps)
for i in xrange(0, l-l_l+1):
sum = 0
plus = 0
while sum < limit and i+plus < l-l_l+1:
sum += ps[i+plus]
plus += 1
if sum < limit and is_prime(sum) and plus > l_l:
l_l = plus
l_p = sum
return l_p
def circular_primes(limit):
result = []
primes = primes_less_than(limit)
primes_dict = defaultdict(bool)
for p in primes:
primes_dict[p] = True
count = 0
for p in primes:
if p in result:
continue
rotations = get_rotations(p)
if all([primes_dict[x] == True for x in rotations]):
for r in rotations:
result.append(r)
return sorted(result)
def circular_primes(limit):
result = []
primes = primes_less_than(limit)
primes_dict = defaultdict(bool)
for p in primes:
primes_dict[p] = True
count = 0
for p in primes:
if p in result:
continue
rotations = get_rotations(p)
if all([primes_dict[x] == True for x in rotations]):
for r in rotations:
result.append(r)
return sorted(result)
def p070():
winning_n = 87109
winning_ratio = 87109.0 / 79180
LIMIT = 10 ** 7
ps = list((p for p in primes_less_than(LIMIT / 11) if p >= 11))
pl = len(ps)
for i in xrange(pl - 1, 0, -1):
x = ps[i]
j, y, n = 0, 11, 11 * x
while n < LIMIT:
t = (x - 1) * (y - 1)
ratio = n * 1.0 / t
if ratio < winning_ratio:
if perms(n, t):
winning_n = n
winning_ratio = ratio
j += 1
y = ps[j]
n = x * y
return winning_n
def tup1_gen():
for p1 in primes():
for p2 in primes_less_than(p1):
if p1 != 2 and p1 != 5 and p2 != 2 and p2 != 5:
if co_concat_prime(p2, p1):
yield (p2,p1)
from primes import primes_less_than
import math
primes = primes_less_than(10000)
def can(odd_number):
for p in primes:
if p > odd_number:
break
diff = odd_number - p
if diff % 2 != 0:
continue
square = diff / 2
x = int(math.sqrt(square))
if x*x == square:
return True
return False
for i in range(35, 99999, 2):
if not can(i):
print i
break
def p037():
truncs = [n for n in primes_less_than(1000000) if truncatable(n)]
return sum(truncs)
def primes_from_to(start, stop):
for p in primes.primes_less_than(stop):
if p >= start:
yield p
import primes
def primes_from_to(start, stop):
for p in primes.primes_less_than(stop):
if p >= start:
yield p
list(primes.primes_less_than(1000000))
known_primes = set(primes.primes.known_primes)
maxp = 0
maxsl = 0
for a in range(len(primes.primes.known_primes)):
for b in range(a-1-maxsl):
s = primes.primes.known_primes[b:a]
if len(s) > maxsl:
if sum(s) in known_primes:
print sum(s), s
maxsl = len(s)
maxp = sum(s)
from primes import primes_less_than
primes = set(primes_less_than(1000000))
print 'starting ..'
def iss(p):
count = 0
#seen = set()
li = list(str(p))
for x in range(10):
for i in range(len(li)):
for j in range(len(li)):
ii = li[i]
jj = li[j]
li[i] = li[j] = str(x)
#print int(''.join(li))
y = int(''.join(li))
#print y
#if x in seen: continue
if y in primes:
#seen.add(x)
#print li
count += 1
li[i] = ii
li[j] = jj
if count == 8:
return True
return False
for p in primes:
if iss(p):
print p
def get_4d_primes():
for p in primes.primes_less_than(10000):
if p > 1000:
yield p
def all_co_good(p1, p2):
if (p1,p2) in acg_known:
return acg_known[(p1,p2)]
val = [p for p in primes_less_than(p2) if p>p1 and co_concat_prime(p1,p) and co_concat_prime(p,p2)]
acg_known[(p1,p2)] = val
return val
import primes primes.primes_less_than(100000000)
def using_generation():
ps = list(primes_less_than(LIMIT+1)) + [LIMIT+1, LIMIT+2]
return sums(1, 1, 0, ps)