def main():
global primes
primes = sieve(10**4)
best = 5
best_n = 1
for p in primes:
for q in primes:
n = p * q
if n > 10**7:
break
phi = (p - 1) * (q - 1)
res = float(n) / phi
if res < best:
if sorted(str(phi)) == sorted(str(n)):
best_n = n
best = res
print best_n, best
def main(): global primes primes = sieve(10 ** 4) best = 5 best_n = 1 for p in primes: for q in primes: n = p * q if n > 10 ** 7: break phi = (p-1) * (q-1) res = float(n) / phi if res < best: if sorted(str(phi)) == sorted(str(n)): best_n = n best = res print best_n, best
cache = {}
from pe_utils import sieve
primes = sieve(10**4)
def solve(n, k=1):
if n == 0:
return 1
if n < 0 or n < k:
return 0
if (n, k) not in cache:
x = 0
for i in primes: #xrange(k, n+1):
if i < k:
continue
if i > n + 1:
break
x += solve(n - i, i)
cache[(n, k)] = x
return cache[(n, k)]
res = 0
i = 0
while res <= 5000:
i += 1
res = solve(i)
from pe_utils import sieve, is_prime
import itertools
primes = sieve(8400)
primes_set = set(sieve(10**6))
set_size = 5
print "got primes"
def merge(a, b):
return int(str(a) + str(b))
def all_primes(chain, p):
for x in chain:
if not is_prime(merge(x, p), primes_set) or not is_prime(
merge(p, x), primes_set):
return False
return True
def make_chain(chain):
if len(chain) == set_size:
return chain
for p in primes:
if p > chain[-1] and all_prime(chain + [p]):
new_chain = make_chain(chain + [p])
if new_chain:
return new_chain
return False
from pe_utils import get_prime_factors, sieve2 as sieve N = 10 ** 6 k = 3 primes = sieve(N) a = get_prime_factors(1, primes) b = get_prime_factors(2, primes) c = get_prime_factors(3, primes) for i in xrange(4, N): d = get_prime_factors(i, primes) #print i, a, b, c if len(set(a)) == len(set(b)) == len(set(c)) == len(set(d)) == 4: print i-3 # starting from 4 ... break a,b,c = b,c,d #print get_prime_factors(8, primes)
from fractions import Fraction as f from pe_utils import sieve max_cycle = 1 max_d = 0 primes = sieve(1000) for d in primes: for c in xrange(1, 1000): x = f(1, d) y = x * (10 ** c) - x z = x * (10 ** c) if z.denominator == 1: # no cycle break if y.denominator == 1: # got cycle if c > max_cycle: max_cycle = c max_d = d break print max_d, max_cycle
l2 = l1[1:-1] add = F(l1[0]) if len(l2) == 1: return add+1/F(l2[0]) return add+F(1)/(F(l2[0])+create_fraction_from_list(l2[1:],0)) def solve_eq(D, m = 1): l = sqrt_to_list(D) l = [l[0]] + l[1:] + l[i:m] f = create_fraction_from_list(l) x, y = f.numerator, f.denominator return x,y powers = set(i**2 for i in xrange(40)) primes = set(sieve(10000)) x = (0, 0) for i in xrange(10000+1): if i in powers or i not in primes: continue s = solve_eq(i) # x = max((s[0], i), x) m = 1 xk = s[0] yk = s[1] n = i if xk**2-n*yk**2 != 1: # fix for the case the result is -1 xk, yk = s[0]*xk+n*s[1]*yk, s[1]*xk+s[0]*yk assert xk**2-n*yk**2 == 1
"""
Distinct primes factors
Find the first four consecutive integers to have four distinct prime factors each. What is the first of these numbers?
"""
from pe_utils import sieve
p = sieve(1000000)
def num_pfactors(n):
cnt = 0
if p[n]:
return 0
x = n
i = 2
while x != 1:
if not p[i]:
i += 1
continue
if x % i == 0:
cnt += 1
while x % i == 0:
x //= i
i += 1
return cnt
nf = dict()
for i in range(647, 1000000):
if num_pfactors(i) != 4:
from math import log
from pe_utils import sieve
N = 50 * 10**6
primes = sieve(int(N**0.5) + 1)
primes2 = sieve(int(N**0.34) + 1)
primes3 = sieve(int(N**0.25) + 1)
print log(len(primes), 2)
print log(N, 2)
t = 0
x = set()
for a in primes:
a_q = a**2
for b in primes2:
#if b > a:
# break
b_q = b**3
for c in primes3:
#if c > b:
# break
y = a_q + b_q + c**4
if y < N:
#print a,b,c, a_q,b_q, a_q+b_q+c**4
t += 1
x.add(y)
else:
break
print len(x)
from pe_utils import sieve def count_digits(n): from math import log return int(log(n, 10)) + 1 primes2 = sieve(10 ** 6) primes = set(primes2) print "ready" digits = range(1, 10, 2) + [2] found = range(1, 10, 2) + [2] def is_trunc(n): bkp_n = n while n > 1: if n not in primes: return False n /= 10 n = bkp_n while n > 1: if n not in primes: return False n = n % 10 ** (count_digits(n)-1) return bkp_n % 10 != 1 and str(bkp_n)[0] != "1"# / 10 ** (count_digits(bkp_n)-1) != 1 return True while len(set(found)) < 11:
from pe_utils import sieve2, get_prime_factors, is_prime, sieve primes = None small_primes = sieve(1000) print "sieve created" def phi_n(n): res = n been = [] for p in set(get_prime_factors(n, primes)): res /= p res *= p-1 return res def old_main(): global primes prime = sieve2(10 ** 7 + 2) best = 5 best_n = 1 LIM = 10**7 for n in xrange(2, LIM+1): if any(not n % x for x in small_primes): # for ratio to be smallest we need phi(n) higher so n musn't divide with lot's of primes :) continue phi = phi_n(n) res = float(n) / phi if res < best: if sorted(str(phi)) == sorted(str(n)): best_n = n best = res
from pe_utils import sieve from math import sqrt N = 30 primes = sieve(int(sqrt(N)))0 print primes
from pe_utils import sieve primes = set(sieve(10 ** 6)) best = 0 best_score = 0 for a in xrange(-1000, 1000): for b in xrange(-1000, 1000): c = 0 for n in xrange(1000): if n ** 2 + a*n+b in primes: c += 1 else: break if c > best_score: best = (a, b) best_score = c print best[0] * best[1]
from fractions import Fraction as f
from pe_utils import sieve
max_cycle = 1
max_d = 0
primes = sieve(1000)
for d in primes:
for c in xrange(1, 1000):
x = f(1, d)
y = x * (10**c) - x
z = x * (10**c)
if z.denominator == 1: # no cycle
break
if y.denominator == 1: # got cycle
if c > max_cycle:
max_cycle = c
max_d = d
break
print max_d, max_cycle
cache = {}
from pe_utils import sieve
primes = sieve(10 ** 4)
def solve(n, k = 1):
if n == 0:
return 1
if n < 0 or n < k:
return 0
if (n,k) not in cache:
x = 0
for i in primes:#xrange(k, n+1):
if i < k:
continue
if i > n+1:
break
x += solve(n-i, i)
cache[(n,k)] = x
return cache[(n, k)]
res = 0
i = 0
while res <= 5000:
i += 1
res = solve(i)
print i, solve(i)
LIM = 10**5
LIM2 = int(LIM**0.5)
odds = set(xrange(3, LIM, 2))
from pe_utils import sieve
primes = sieve(LIM)
my_nums = set()
for p in primes:
for i in xrange(LIM2):
my_nums.add(p + 2 * i**2)
print sorted(list(odds - my_nums))[0]
from pe_utils import sieve
from itertools import combinations
N = 10**6
primes = sieve(N)
primes_set = set(primes)
def digit_len(n):
return len(str(n))
def change_num_dig(num, dig, val):
num2 = num
for i in xrange(1, dig):
num2 /= 10
cur_val = num2 % 10
dif = cur_val - val
num = num - dif * pow(10, dig - 1)
return num
def apply_mask(n, comb, val, best_c=10):
p2 = n
for x in comb:
p2 = change_num_dig(p2, x + 1, val)
return p2
def prime_digit_replace(p, comb, best_c=10):
c = 0
from pe_utils import sieve2, get_prime_factors, is_prime, sieve
primes = None
small_primes = sieve(1000)
print "sieve created"
def phi_n(n):
res = n
been = []
for p in set(get_prime_factors(n, primes)):
res /= p
res *= p - 1
return res
def old_main():
global primes
prime = sieve2(10**7 + 2)
best = 5
best_n = 1
LIM = 10**7
for n in xrange(2, LIM + 1):
if any(
not n % x for x in small_primes
): # for ratio to be smallest we need phi(n) higher so n musn't divide with lot's of primes :)
continue
phi = phi_n(n)
res = float(n) / phi
if res < best: