def prime_powers(n):
''' Return dict where keys are primes and values are exponents. Represents
prime factorization of 'n!' '''
powers = dict()
for p in primes(n):
powers[p] = legendre(p, n)
return powers
def longest_recurring(n):
for d in primes(n)[::-1]:
p = 1
while pow(10, p, d) != 1:
p += 1
if d - 1 == p:
return d
def main():
''' Driver function '''
n, m = 20 * 10**6, 15 * 10**6
total = 0
for p in primes(n):
total += p * (legendre(n, p) - legendre(m, p) - legendre(n - m, p))
print(total)
def getSolution(limit):
result = 1
for prime in pyprimesieve.primes(limit):
power = 1
while prime ** (power + 1) <= limit:
power += 1
result *= prime ** power
return result
def main():
''' Driver function '''
end = 40000000
chain_lengths = [0] * end
total = 0
for p in primes(end):
if prime_chain_length(p, chain_lengths) == 25:
total += p
print(total)
def path_to_roots(bitmask):
''' Generator of lists containing the values between 'p' and its digital
root. Values are represented using 'bitmask' '''
for p in primes(10**7, 2 * 10**7):
cur = []
while p > 9:
digits = [int(n) for n in str(p)]
cur += [[bitmask[d] for d in digits]]
p = sum(digits)
yield cur + [[bitmask[p]]]
def main():
''' R(k) is equal to (10**k-1)/9. Thus, a number n divides R(k) IFF
(10**k-1)/9 = 0 (mod n). This can be re-arranged to 10**k = 1 (mod 9*n),
which allows for the use of modular exponentiation '''
factors = []
for n in primes(10**6):
if pow(10, 10**9, 9*n) == 1:
factors += [n]
if len(factors) == 40:
break
print(sum(factors))
def maximise_quad():
max_seq = A = B = 0
quad = lambda n, a, b: n * n + a * n + b
cache = set(pp.primes(100000))
for a in range(-999, 1000, 1):
for b in range(-1000, 1001, 1):
n = 0
while (x := quad(n, a, b)) in cache:
n += 1
if n > max_seq:
A, B, max_seq = a, b, n
def get_vals(end):
''' This post explains it: https://bit.ly/2J9ByKf '''
vals = primes(end)
n = 0
while vals[n] < end**0.5:
val, l = 0, 1
while val < end:
val = vals[n]**2**l
vals.append(val)
l += 1
n += 1
return sorted(vals)
def find_num_divisors(end):
''' Calculate the number of divisors for each number up to 'end' '''
num_divisors = [0 for _ in range(2, end)]
prime = set(primes(10**7))
for n in range(2, end):
divisors = 1
if n not in prime:
factors = factorize(n)
for _, e in factors:
divisors *= (e + 1)
num_divisors[n - 2] = divisors
return num_divisors
def main():
''' The solution must be > 1000000 as A(n) < n. This function iterates over
valid values of 'n' (not prime and not divisible by 2 or 5) starting at 2
until 25 solutions are found. The sum of these solutions is then printed '''
found = []
prime = set(primes(100000))
for n in count(2):
if n not in prime and n % 2 and n % 5 and (n -
1) % brute_check(n) == 0:
found += [n]
if len(found) == 25:
break
print(sum(found))
def main():
''' 'n' must be a cube. This function iterates over all cubes up to 600^3
for each prime. 'found' keeps track of the number of valid '(n, p)' pairs.
This amount is printed at the end. This solution is relatively slow (about
30 seconds) '''
found = 0
cubes = [n**3 for n in range(1, 600)]
for p in primes(1000000):
for n in cubes:
cur = n * n * (n + p)
if round(cur**(1 / 3))**3 == cur:
found += 1
break
print(found)
def S(n):
''' Uses Goldbach's conjecture, which has been verified for integers in the
range required by the problem. The sum of all P(m, 1) for 1 <= m <= n is
the number of primes <= n. The sum of all P(m, 2) is the number of even
integers in 4 <= m <= n plus the number of integers in 6 <= m <= n equal to
a prime plus 2. The sum of all P(m, k) with k >= 3 is the sum of all
integers in 6 <= m <= n floor divided by 2 minus 2 (subtract 2 for each m
or subtract 2*(n-6) once). 'P_n_k' is a closed form expression that
calculates this '''
prime = primes(n + 1)
P_n_1 = len(prime)
P_n_2 = 0 if n < 4 else (n // 2) - 1 + (prime[-1] + 2 <= n) + P_n_1 - 2
P_n_k = 0 if n < 6 else (ceil((7 - n) / 2)**2 - 7 * ceil(
(7 - n) / 2) + (n // 2)**2 + (n // 2) - 4 * n + 20) // 2
return P_n_1 + P_n_2 + P_n_k
def general_hamming(end, high):
''' Return a set of all numbers (<= end) with prime factors not exceeding
high '''
hamming = {1}
prev = {1}
base = primes(high+1)
while 2*min(prev) <= end:
cur = set()
for n in prev:
for p in base:
temp = n*p
if temp > end:
break
cur.add(temp)
hamming.update(cur)
prev = cur
return hamming
def find_goldbach():
primes = pp.primes(N)
cache = set(primes)
sq2 = [2 * (n * n) for n in range(1, N)]
i = 9
while (i < N):
if not i in cache:
lt = lambda x: x < i
result = False
for a in takewhile(lt, sq2):
for b in takewhile(lt, primes):
if a + b == i:
result = True
break
if result:
break
if not result:
return i
i += 2
def test_1(self):
self.assertEqual(pyprimesieve.primes(1), [])
def pyprimesieve(n):
import pyprimesieve
return list(pyprimesieve.primes(n))
def sumofprimes(n): # lambda expression is slower
return sum(primes(n))
#! /usr/bin/env python3
# -*- coding: utf-8 -*-
# projecteuler.net
#
# problem 41
#
import pyprimesieve
primes = pyprimesieve.primes(7654321)
def pandigital(n):
s = str(n)
if ''.join(sorted(s)) == "1234567":
return True
return False
for p in reversed(primes):
if pandigital(p):
print(p)
exit(0)
def test_emptylist_2(self):
self.assertEqual(pyprimesieve.primes(-1), [])
def test_prime_count_2(self):
self.assertEqual(len(pyprimesieve.primes(10**8)), 5761455)
def test_prime_count_1(self):
self.assertEqual(len(pyprimesieve.primes(10**7)), 664579)
def test_prime_count_2(self):
self.assertEqual(len(pyprimesieve.primes(10**8)), 5761455)
def test_4(self):
self.assertEqual(pyprimesieve.primes(4), [2, 3])
def test_3(self):
self.assertEqual(pyprimesieve.primes(3), [2])
def test_2(self):
self.assertEqual(pyprimesieve.primes(2), [])
def test_2(self):
self.assertEqual(pyprimesieve.primes(2), [])
def test_ranges_1(self):
# start > n
self.assertEqual(pyprimesieve.primes(411, 42), [])
def test_4(self):
self.assertEqual(pyprimesieve.primes(4), [2, 3])
def test_ranges_2(self):
# start < 0 - should be uneffected as if it were 2
self.assertTrue(sequences_equal(pyprimesieve.primes(-100, 42), primes(42)))
def test_bignums_2(self):
self.assertEqual(pyprimesieve.primes_sum(10**6), sum(pyprimesieve.primes(10**6)))
def test_ranges_3(self):
# arbitrary point to start
self.assertTrue(sequences_equal(pyprimesieve.primes(1412, 85747),
dropwhile(lambda n: n < 1412, primes(85747))))
def sumofprimes(n): # lambda expression is slower
return sum(primes(n))
def test_ranges_4(self):
# arbitrary point to start
self.assertTrue(sequences_equal(pyprimesieve.primes(74651, 975145),
dropwhile(lambda n: n < 74651, primes(975145))))
result = 0
for num in droots:
result += 2*aroot(num)
return result
def maxclock(n):
n_copy = str(n)
droots = [n_copy]
while int(n_copy) >= 10:
n_copy = droot(n_copy)
droots.append(n_copy)
result = aroot(droots[0])
for i, num in enumerate(droots[1:]):
prev = droots[i]
cur = num
diff = len(prev) - len(cur)
result += aroot(prev[:diff])
for i, dig in enumerate(cur):
result += chdict[(prev[diff+i], dig)]
result += aroot(droots[-1])
return result
import pyprimesieve as pp
result = 0
for p in pp.primes(2*10**7):
if p>10**7:
s = samclock(p)
m = maxclock(p)
result += (s-m)
print result
from math import log
import pyprimesieve as pp
import sys
sys.setrecursionlimit(5000)
def max_primepow(n, prime):
if n < prime:
return 0
return int(log(n)/log(prime))
myprimes = set(pp.primes(10))
def gen_admissible(n, prodprimes=1, lastprime=1, admiss=[1]):
nextprime = None
for num in xrange(lastprime+1, 6):
if num in {2, 3, 5}:
nextprime = num
prodprimes = prodprimes * nextprime
break
if not nextprime:
return admiss
else:
newadmiss = []
for base in sorted(admiss):
max_pow = max_primepow(n/base, nextprime)
if not max_pow:
break
newadmiss += [base*nextprime**i for i in xrange(1, max_pow+1)]
import pyprimesieve as pp
import sys
from itertools import product
from operator import mul
sys.setrecursionlimit(2000)
gazinga_cache = {1: 1}
prime = pp.primes(45)
def isvalid(combo, nmax=10**9):
num = 1
for i, n in enumerate(combo):
num = num * prime[i]**n
return num < nmax, num
def firstidx(combo):
idx = 0
for i in xrange(len(combo)-1, 0, -1):
if combo[i] < combo[i-1]:
idx = i
break
combo[idx] += 1
newcombo = combo[:idx+1] + [1]*len(combo[idx+1:])
return idx, newcombo
def secondidx(combo, nmax=10**9):
def prime_sieve(n):
"""
Finds all primes from 1 until n
"""
import pyprimesieve
return pyprimesieve.primes(int(n))
#! /usr/bin/env python
# -*- coding: utf-8 -*-
# projecteuler.net
#
# problem 27
#
from pyprimesieve import primes
s = 0
c = (0, 0)
p = set(primes(751000))
for a in range(-999, 0, 2):
for b in range(-a, 1000, 2):
n = 1
while (n * (n + a) + b) in p:
n += 1
if n > s:
s = n
c = (a, b)
print(c[0] * c[1])
sequences_equal(pyprimesieve.primes(-100, 42), primes(42)))
def test_ranges_3(self):
# arbitrary point to start
self.assertTrue(
sequences_equal(
pyprimesieve.primes(1412, 85747),
itertools.dropwhile(lambda n: n < 1412, primes(85747))))
def test_ranges_4(self):
# arbitrary point to start
self.assertTrue(
sequences_equal(
pyprimesieve.primes(74651, 975145),
itertools.dropwhile(lambda n: n < 74651, primes(975145))))
def sequences_equal(lst1, lst2):
return all(a == b for a, b in zip_longest(lst1, lst2))
for i, n in enumerate(
xrange(100, 10000, 100)
): # create sequence comparison tests for sieves of size n in range
test = lambda self: self.assertTrue(
sequences_equal(pyprimesieve.primes(n), primes(n)))
setattr(TestPrimes, 'test_' + str(i), test)
if __name__ == "__main__":
unittest.main()
def test_ranges_1(self):
# start > n
self.assertEqual(pyprimesieve.primes(411, 42), [])
# start > n
self.assertEqual(pyprimesieve.primes(411, 42), [])
def test_ranges_2(self):
# start < 0 - should be uneffected as if it were 2
self.assertTrue(sequences_equal(pyprimesieve.primes(-100, 42), primes(42)))
def test_ranges_3(self):
# arbitrary point to start
self.assertTrue(sequences_equal(pyprimesieve.primes(1412, 85747),
dropwhile(lambda n: n < 1412, primes(85747))))
def test_ranges_4(self):
# arbitrary point to start
self.assertTrue(sequences_equal(pyprimesieve.primes(74651, 975145),
dropwhile(lambda n: n < 74651, primes(975145))))
def sequences_equal(lst1, lst2):
return all(a == b for a, b in izip_longest(lst1, lst2))
primes = _primes_numpy if HAS_NUMPY else _primes
for i, n in enumerate(xrange(100, 10000, 100)): # create sequence comparison tests for sieves of size n in range
test = lambda self: self.assertTrue(sequences_equal(pyprimesieve.primes(n), primes(n)))
setattr(TestPrimes, 'test_' + str(i), test)
if __name__ == "__main__":
unittest.main()
from math import sqrt, log
import pyprimesieve as pp
import sys
sys.setrecursionlimit(5000)
def max_primepow(n, prime):
if n < prime:
return 0
return int(log(n)/log(prime))
myprimes = set(pp.primes(10**9))
def gen_admissible(n, prodprimes=1, lastprime=1, admiss=[1], valids=[1]):
nextprime = None
for num in xrange(lastprime+1, int(sqrt(n))+1):
if num in myprimes:
nextprime = num
prodprimes = prodprimes * nextprime
break
if not nextprime:
return admiss
else:
newadmiss = []
for base in sorted(valids):
max_pow = max_primepow(n/base, nextprime)
if not max_pow:
break
newadmiss += [base*nextprime**i for i in xrange(1, max_pow+1)]
def test_ranges_3(self):
# arbitrary point to start
self.assertTrue(
sequences_equal(
pyprimesieve.primes(1412, 85747),
itertools.dropwhile(lambda n: n < 1412, primes(85747))))
def test_1(self):
self.assertEqual(pyprimesieve.primes(1), [])
def test_ranges_2(self):
s = pyprimesieve.primes_sum(7, 22)
self.assertEqual(s, 67)
self.assertEqual(s, sum(pyprimesieve.primes(22)[3:]))
def test_3(self):
self.assertEqual(pyprimesieve.primes(3), [2])
def test_ranges_1(self):
s = pyprimesieve.primes_sum(3, 13)
self.assertEqual(s, 26)
self.assertEqual(s, sum(pyprimesieve.primes(13)[1:]))
def test_prime_count_1(self):
self.assertEqual(len(pyprimesieve.primes(10**7)), 664579)
import pyprimesieve as pp
from benchmark import timed
max_prime = 1000000
primes = pp.primes(max_prime)
max_idx = len(primes) - 1
cache = set(primes)
# optimize:
# work from max slice down (and return first result), not min slice up
@timed
def find_longest_slice(min_slice):
result = 0
longest_slice = min_slice
for idx in range(0, max_idx - min_slice):
slice = min_slice
while (subset := sum(primes[idx:idx + slice])) <= max_prime:
if subset in cache:
if slice > longest_slice:
longest_slice = slice
result = subset
slice += 1
return (result, longest_slice, idx)
result = find_longest_slice(21)
print(result)
def test_emptylist_2(self):
self.assertEqual(pyprimesieve.primes(-1), [])
import pyprimesieve as pp
primes = set(pp.primes(500))
from collections import defaultdict
from fractions import Fraction as F
def shoutprob(yelp, k):
res = F(1, 3) * (F(1) + F(k in primes))
if yelp == 'P':
return res
else:
return F(1) - res
def getjumps(k, n=500):
if k == 1:
return {k+1: F(1)}
elif k == n:
return {k-1: F(1)}
else:
return {k-1: F(1, 2), k+1: F(1, 2)}
def movefrog(yelp, possib):
newpossib = defaultdict(F)
for k, prob in possib.iteritems():
for j, jprob in getjumps(k).iteritems():
newpossib[j] += prob*shoutprob(yelp, k)*jprob
return newpossib
def test_ranges_2(self):
# start < 0 - should be uneffected as if it were 2
self.assertTrue(
sequences_equal(pyprimesieve.primes(-100, 42), primes(42)))
import pyprimesieve
# Library from here: https://pypi.python.org/pypi/pyprimesieve
first_number = "1"+"0"*30+"1"
last_number = "1"+"1"*30+"1"
first_thousand_primes = pyprimesieve.primes(5000)
def is_prime_in_base(num_str, base):
n = int(num_str, base=base)
for i in first_thousand_primes:
if n %i ==0: return i
return -1
print "Case #1:"
J = 0
current_number = first_number
while J!=500:
factors = []
correct = True
for base in xrange(2,10+1): # for each base
factor = is_prime_in_base(current_number, base)
if factor != -1:
factors.append(factor)
else:
correct = False
break
if correct:
print current_number,
for factor in factors: print factor,
print ""
J+=1
def test_ranges_4(self):
# arbitrary point to start
self.assertTrue(
sequences_equal(
pyprimesieve.primes(74651, 975145),
itertools.dropwhile(lambda n: n < 74651, primes(975145))))
import pyprimesieve as pp
from collections import defaultdict
primelst = pp.primes(10**7)
myprimes = set(primelst)
def isprime(n):
return n in myprimes
def changedigit(n):
str_n = str(n)
edges = defaultdict(set)
for i, dig in enumerate(str_n):
alldigs = [str(x) for x in range(10) if str(x) != dig]
if i == 0:
alldigs.pop(0)
for newdig in alldigs:
testprime = str_n[:i] + newdig + str_n[i+1:]
testprime = int(testprime)
if isprime(testprime):
edges[testprime].add(n)
return edges
def adddigit(n):
str_n = str(n)
edges = defaultdict(set)
alldigs = [str(x) for x in range(1, 10)]
for newdig in alldigs:
testprime = newdig + str_n