def test_max_prime_overlaps_interval():
# Test when max_prime > N
for s, g, mp, expected in (
(1, 99, 2, 50),
(1, 99, 3, 34),
(1, 99, 5, 28),
(1, 99, 7, 25),
(1, 99, 11, 25),
(1, 99, 100, 25),
(1, 99, 200, 25),
):
#print (s, g, mp, expected, [s+i for i, v in enumerate(utils.sieve(s, g, mp)) if not v])
assert expected == utils.sieve(s, g, mp).count(False)
assert utils.sieve(s, g, mp) == brute(s, g, mp)
def main():
sums = []
prime_sum = 0
primes = [p for p in sieve(LIMIT)]
for p in primes:
prime_sum += p
sums += [prime_sum]
maxlen = 0
number = 0
for index, s in enumerate(sums):
if s < LIMIT and in_primes(primes, s):
maxlen = index
number = s
for i in range(index - maxlen - 1, -1, -1):
sr = s - sums[i]
if sr >= LIMIT:
break
if in_primes(primes, sr):
maxlen = index - i
number = sr
print("Prime =", number, "Longest chain =", maxlen)
def test_limit_limitted():
# Test that limit is decreased to sqrt(s + g)
mp = 10 ** 10
for s, g, expected in (
(1, 99, 25),
(1000, 1000, 135),
(10 ** 6, 100, 6),
(10 ** 9, 100, 7),
):
composites = utils.sieve(s, g, mp)
for i, c in enumerate(composites):
assert gmpy2.is_prime(s + i) == (not c)
assert expected == composites.count(False)
def test_sieve_primepi():
# Easy to generate these with `primesieve <start> <start + gap>
for s, g, mp, expected in (
(0, 100, 10, 25),
(1, 99, 10, 25),
(1, 99, 100, 25),
(101, 100, 20, 21),
(101, 100, 100, 21),
(1001, 1000, 3, 334),
(1001, 1000, 30, 151),
(1001, 1000, 50, 135),
(1000001, 10000, 1100, 753),
):
assert expected == utils.sieve(s, g, mp).count(False)
def test_zero_one_composite():
# Test that 0, 1 are "composite" in our sieve
for s, g in (
(0, 10),
(1, 5),
(0, 1),
(0, 2),
(1, 1),
(1, 2)
):
composites = utils.sieve(s, g, 10)
if s == 0:
assert composites[0] == True
if s + g - 1 >= 1:
assert composites[1 - s] == True
def loop(absupto):
""" We know that b must be prime and a must be odd
"""
from utils import sieve
possbs = sieve(absupto)
max_consec = 0
consec_ab = (0,0)
for b in possbs:
for a in range(-b, absupto, 2):
length = len(list(fns(a,b)))
if length > max_consec:
max_consec = length
consec_ab = (a, b)
return max_consec, consec_ab
def loop(absupto):
""" We know that b must be prime and a must be odd
"""
from utils import sieve
possbs = sieve(absupto)
max_consec = 0
consec_ab = (0, 0)
for b in possbs:
for a in range(-b, absupto, 2):
length = len(list(fns(a, b)))
if length > max_consec:
max_consec = length
consec_ab = (a, b)
return max_consec, consec_ab
def test_sieve():
for s, g, mp in (
(1, 100, 10),
(11, 89, 10),
(100, 100, 10),
(1001, 100, 10),
(1001, 1000, 10),
(1001, 100, 50),
(1001, 1000, 50),
(5, 1000, 3),
):
expect = brute(s, g, mp)
result = utils.sieve(s, g, mp)
assert len(expect) == len(result)
#print([1 * v for v in expect])
#print([1 * v for v in result])
#print([s + i for i, v in enumerate(expect) if v])
print([(s+i, a, b) for i, (a, b) in enumerate(zip(expect, result)) if a != b])
assert expect == result, (s, g, mp)
# Pandigital prime
# n != 8, 9
from utils import sieve
primes = sieve(8000000)
def check(n):
n = str(n)
l = len(n)
if "0" in n:
return False
for i in range(1, l+1):
if str(i) not in n:
return False
return True
candidate = [p for p in primes if check(p)]
print candidate[-1]
import utils
prime_list = utils.sieve(1e6)
def check_devided(p):
"""
"""
power_of_power = 9
mod = 10
for i in range(power_of_power):
mod = mod**10 % p
if mod % p == 1:
return True
else:
return False
result = 0
count = 0
for p in prime_list[2:]: #because it not devided by 2 and 3
check = check_devided(p)
if check:
result += p
count += 1
if count == 40:
print(result)
# Problem 49
# real 0m0.153s
# user 0m0.125s
# sys 0m0.015s
from utils import sieve
from itertools import permutations
from utils import is_prime
primes = {p for p in sieve(10000) if len(str(p)) == 4}
def solve():
for p in primes:
# something something permutations
digits = [digit for digit in str(p)]
nums = []
for perm in permutations(digits):
nums.append(''.join(str(d) for d in perm))
# Ok, so now I have all the 4 bit rotated numbers.
solution_nums = {int(num) for num in nums if int(num) in primes}
if(len(solution_nums) <4): continue
for n in solution_nums:
if (n-p == 3330 and n+3330 in solution_nums):
return(p,n,n+3330)
print(solve())
# Problem 35
# real 0m0.818s
# user 0m0.769s
# sys 0m0.038s
from utils import is_prime
from utils import sieve
primes = [p for p in sieve(1000000)]
def is_rotatable(p):
for a in range(1,len(p)):
if not is_prime(int(p[a:]+p[:a])):
return False
return True
rotatable_primes = [p for p in primes if is_rotatable(str(p))]
print(len(rotatable_primes))
# Largest prime factor
from utils import sieve
from math import sqrt
target = 600851475143
primes = sieve(int(target))
p_index = 0
target_factor = []
while True:
p = primes[p_index]
if target < p:
break
elif target % p == 0:
target /= p
target_factor.append(p)
else:
p_index += 1
print target_factor
#!/usr/bin/env python
from utils import sieve
size = 2000000
prime = sieve(size)
i = 0
sum = 0
while i < size:
if prime(i): sum = sum + i
i = i + 1
print sum
#!/usr/bin/env python
import sys
from collections import Counter
from utils import sieve
P = 47
isprime = sieve(P)
primes = []
for i in xrange(P + 1):
if isprime[i]:
primes.append(i)
# it's same as for up to 10^14
# so that's how we know when to stop :)
# technically supposed to use:
# https://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem
# to get the bound.
# w/e, I'm too busy to know random number theory facts.
MAX = 10**13
def printc(c):
todelete = []
for k, v in c.iteritems():
if v == 0:
todelete.append(k)
for k in todelete:
del c[k]
psmooth = set()
# Truncatable primes
from utils import sieve
limit = 1000000
primes = sieve(limit)
primes_set = set(primes)
def check(p):
p = str(p)
n = len(p)
if n == 1:
return False
for s in range(1, n):
if int(p[s:]) not in primes_set:
return False
if int(p[:s]) not in primes_set:
return False
return True
candidate = [p for p in primes if check(p)]
print(sum(candidate))
# -*- coding: utf-8 -*-
"""
Created on Mon Apr 16 14:55:47 2018
@author: lamwa
"""
import utils
RANGE = 1000000
_, p = utils.sieve(RANGE)
n = 1
while True:
mod = p[n - 1] * p[n - 1]
rem = (utils.power_mod(p[n - 1] - 1, n, mod) +
utils.power_mod(p[n - 1] + 1, n, mod)) % mod
if rem >= 10**10:
break
n += 1
print(n)
# Quadratic primes
# n2 + an + b
# b is prime
from utils import sieve
primes = sieve(3000000)
primes_set = set(primes)
def check_prime_chain(a, b):
n = 0
while True:
k = n**2 + a * n + b
if k in primes_set:
n += 1
else:
break
return n
max_a, max_b = 0, 0
max_chain = 0
# b = primes < 1000
possible_b = primes[:168] + [-i for i in primes[:168]]
for a in range(-1000 + 1, 1000):
for b in possible_b:
c = check_prime_chain(a, b)
if c > max_chain:
max_chain = c
def notdecomposeable(odd):
return not any(sqrt((odd - pr) / 2.).is_integer() for pr in sieve(odd))
def notdecomposeable(odd):
return not any(sqrt((odd - pr)/2.).is_integer() for pr in sieve(odd))
achieve same # of factors in a and b by keeping track of the largest
multiplicity possible for each prime for factorizing multiplicities starting
from the ith factor of N!. Note that we get the most pruning by considering the
factors of N! in decreasing order of multiplicity.
Last trick is a counting trick: if a<b and a and b have same number of factors,
note that we will double count this case if we do not have some extra
bookkeeping. Instead of tracking whether a<b in the dp (which is hard), we
track whether a==b in the dp (which is much easier), and if the have the same
number of factors in the end, we add 2 to the total instead of 1. Since we have
now double-counted all the cases, the final answer is achieved halving.
"""
N = 100
# we need to factor largest multiplicity of anything <= 100 (+1), which will never exceed 100 itself
primes = np.nonzero(sieve(N))[0]
factors = Counter()
for i in xrange(2, N+1):
factors.update(factor(i, primes))
_mem = {}
def same_divisor_count(i, div_factor_diff, equal, factorcounts, bounds):
# remove 0 entries and do bounds check
for k, c in div_factor_diff.items():
if c==0: del div_factor_diff[k]
else:
if i==len(bounds) or abs(c) > bounds[i][k]:
return 0
if i==len(factorcounts):
assert len(div_factor_diff) == 0
return 2 if equal else 1
def main():
primes = utils.sieve(10000)
composites = set(n for n in range(2, 10000) if n not in primes)
twicesquares = set(2*(n**2) for n in range(100))
sums = set(sum(c) for c in product(primes, twicesquares))
print(min(n for n in composites if n not in sums and n % 2 != 0))
from utils import sieve
from itertools import permutations, product
import re
primes = sieve(1000000)
primes_set = set(primes)
def generate_number(template):
a = re.sub("\*", "0", template)
b = re.sub("\d*(.*)", "\\1", template)
b = re.sub("\d", "0", b)
b = re.sub("\*", "1", b)
# return a,b
a, b = int(a), int(b)
number = [a + b * n for n in range(10)]
if template[0] == "*":
number = number[1:]
return number
def count_primes(number):
np = 0
for n in number:
if n in primes_set:
np += 1
return np
def generate_template(nstar, ndigit):
templates = []
samClockCountDict = {}
def samClockCount(n):
if n in digitPatterns:
return 2*len(digitPatterns[n])
elif n not in samClockCountDict:
cost = sum(len(digitPatterns[int(digit)]) for digit in str(n))
samClockCountDict[n] = cost*2 + samClockCount(digitalRootStep(n))
return samClockCountDict[n]
#maxClockCountDict = {}
#def maxClockCount(n):
#for n in range(a,b):
if __name__ == "__main__":
a = 10**7
b = 2 * a
_primes = sieve(b)
for index in range(len(_primes)):
if _primes[index] > a:
_primes = _primes[index:]
break
print(_primes)
for p in _primes:
print(p,samClockCount(p))
#!/usr/bin/env python
import utils
MAX = 100 + 1
prime = utils.sieve(MAX)
def prime_factors(n):
factors = []
while not prime(n):
for i in range(n):
if prime(i) and n % i == 0:
factors.append(i)
n = n / i
break
factors.append(n)
return factors
# prime factor each a/b
factors = [[], []]
for i in range(2, MAX):
factors.append(prime_factors(i))
def encode(factors, exponent):
factors = factors[:]
encoded = ""
while factors:
expsum = exponent
base = factors.pop(0)
#!/usr/bin/env python
import utils
prime = utils.sieve(1200000)
quad = lambda a, b, n: (n**2) + (a*n) + b
def consec_primes(a, b):
n = 0
while True:
if prime(quad(a, b, n)):
n = n + 1
else:
return n
largest = (1, 1, 1)
for a in xrange(-1000, 1000):
for b in xrange(-1000, 1000):
if b % 2 == 0 or\
b % 3 == 0 or\
b % 5 == 0:
pass
else:
prime_count = consec_primes(a, b)
if prime_count > largest[2]:
largest = (a, b, prime_count)
print largest[0] * largest[1]
""" Project Euler Problem 10 ======================== The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17. Find the sum of all the primes below two million. """ from utils import sieve MAX = 2000000 print sum(sieve(MAX))
#Problem 50
# real 0m1.834s
# user 0m1.765s
# sys 0m0.042s
from utils import sieve
from utils import is_prime
primes = sieve(1000000)
max_primes = 1
for x in range(78498-50):
for y in range(499,600):
total = sum(primes[x+z] for z in range(y))
if total in primes and y > max_primes:
print(y,total)
max_primes = y
print(max_primes)
# -*- coding: utf-8 -*-
"""
Created on Sun Apr 15 11:26:43 2018
@author: lamwa
"""
import utils
RANGE = 100
TH = 4000000
_, p_list = utils.sieve(RANGE)
ans = 1e20
def dfs(n, i, cur, r):
global ans
if n >= ans:
return None
if (cur + 1) // 2 > TH:
ans = n
return None
for t in range(1, r + 1):
n *= p_list[i]
dfs(n, i + 1, cur * (2 * t + 1), t)
dfs(1, 0, 1, 100)
from utils import sieve
from math import sqrt
l = sieve(100000)
def f():
n = 1
while (1>0):
for x in l:
if (x >=n): break;
print n,x
a = sqrt((n-x)%2)
if a == int(a): continue
else: return n
n +=2
def g():
for x in xrange(10,20):
for y in l:
if y >= x: break
print x,y,x-y
print f()
# Problem 10 # real 0m0.389s # user 0m0.318s # sys 0m0.062s from utils import sieve print(sum(sieve(2000000)))
from utils import sieve
from math import log, sqrt
import time
N = 10000000
primes = sieve(N / 2)
def M(p1, p2, N):
N1 = int(log(N, p1))
N2 = int(log(N, p2))
candidate = [
p1**i * p2**j for i in range(1, N1 + 1) for j in range(1, N2 + 1)
]
max_candidate = 0
for c in candidate:
if c <= N and c > max_candidate:
max_candidate = c
return max_candidate
start = time.time()
sum_M = 0
max_i = sqrt(N)
for i in primes:
if i <= max_i:
max_j = N / i
for j in primes:
if j <= max_j and j > i:
m = M(i, j, N)
sum_M += m
#!/usr/bin/env python
import tqdm
from utils import c, sieve
"""
Idea: write S(L) as a summation, sum over terms with
same power of 5. Tricky part is computing the coefficient;
for this we can use identity
sum(c(n,i) for i in xrange(k+1)) == c(n+1,k+1)
"""
N = 50*10**6
prime = sieve(N)
pi = [0]*(N+1)
for i in xrange(2,N+1):
pi[i] = pi[i-1] + prime[i]
def check(n):
ret = 0
for k in xrange(pi[n]+1):
ret += 6*c(n-1,k)*5**(n-1-k)
return ret
mod = 10**9+7
ans=0
stop = 0
for diff in tqdm.tqdm(xrange(1, N+1)):
while stop<N and stop+1-pi[stop+1]<=diff:
stop += 1
# 10001st prime # pn ~ n/log(n) # 200000./log(200000) = 16385 from utils import sieve from math import log n = 200000 primes = sieve(n) print len(primes) print primes[10000]
import utils
RANGE = 20000
def A(n):
x = 1
k = 1
while x > 0:
x = (x * 10 + 1) % n
k += 1
return k
is_p, _ = utils.sieve(RANGE)
ans = 0
n = 3
cnt = 0
while True:
if not is_p[n]:
if (n - 1) % A(n) == 0:
ans += n
cnt += 1
if cnt == 25:
break
n += 2
if n % 5 == 0:
n += 2
# Problem 3 # real 0m0.192s # user 0m0.153s # sys 0m0.029s from utils import sieve import math target = 600851475143 primes = sieve(int(math.sqrt(target))) print(max(p for p in primes if not target%p))
from utils import sieve, permuta, lintstr
from math import sqrt
l = sieve(int(sqrt(7654321)))
a = permuta("7654321")
a.reverse()
def check(x):
for y in l:
if x%y == 0: return 0
return 1
def f():
for x in a:
if check(int(x))==1: return x
return 0
print f()
# Problem 87
#real 0m2.592s
#user 0m2.489s
#sys 0m0.071s
from utils import sieve
target = 50000000
quads = sieve(int(target ** (1/4)))
triples = sieve(int(target ** (1/3)))
doubles = sieve(int(target ** (1/2)))
vals = set()
for q in quads:
for t in triples:
for d in doubles:
s = d**2 + t**3 + q**4
if s < target:
vals.add(s)
print(len(vals))