def prime_factors(n):
primes = euler_tools.sieve(n)
factors = collections.defaultdict(list)
for p in primes:
j = 1
while True:
factors[p * j].append(p)
j += 1
if p * j > n:
break
return factors
import euler_tools
import copy
primes = euler_tools.sieve(100)
def n(factorization):
_n = 1
for k, v in factorization.items():
_n *= (k**v)
return _n
def d(factorization):
_d = 1
for k, v in factorization.items():
_d *= (2 * v + 1)
return _d
def replace(factorization, factor, limit):
_factorization = copy.deepcopy(factorization)
for i in range(2, factor):
factorization[factor] = 0
pds = euler_tools.prime_divisors(i)
for pd in pds:
factorization[pd] += 1
print(factor, i, pds, factorization, d(factorization),
d(_factorization))
import euler_tools
N = 1e6
primes = euler_tools.sieve(int(N))
print(primes)
res = []
for p in range(1, int(N) + 1):
k = p + 1
#print(p, k)
_p = k**2 + k * p + p**2
if _p in primes:
res.append(_p)
print(_p, p**3, k * p**2, p, k)
## k = p-1
## _p = k**2+k*p+p**2
## if _p in primes:
## res.append(_p)
## print(_p, p**3, k*p**2, p, k)
if _p > N:
break
with euler_tools.Watch():
print(len(set(res)))
import euler_tools
PRIMES = euler_tools.sieve(190)
FACT = {}
def fact(n):
try:
return FACT[n]
except KeyError:
if n == 0 or n == 1:
f = 1
else:
f = n * fact(n - 1)
FACT[n] = f
return f
def combin(n, k):
return int(fact(n) / (fact(k) * fact(n - k)))
def product(x):
p = 1
for _x in x:
p *= _x
return p
def num_divisors(num_primes):
import euler_tools
N = int(1e5)
primes = euler_tools.sieve(N)
def rads(n):
_rads = [1] * n
for p in primes:
k = 1
while True:
if p * k <= n:
_rads[p * k - 1] *= p
k += 1
else:
break
# print(max(_rads))
_rads = [(n + 1, rad) for n, rad in enumerate(_rads)]
#print(_rads)
return sorted(_rads, key=lambda x: x[1])
from euler_tools import sieve
PRIMES = sieve(100)
print(PRIMES)
def divisible_by_four_primes(n):
nums = []
for i in range(len(PRIMES)):
for j in range(i+1, len(PRIMES)):
for k in range(j+1, len(PRIMES)):
for m in range(k+1, len(PRIMES)):
q = 1
while True:
num = q*PRIMES[i]*PRIMES[j]*PRIMES[k]*PRIMES[m]
print(n/num)
if num < n:
nums.append(num)
q += 1
else:
break
return len(nums), nums
print(divisible_by_four_primes(10000000000000000))
import euler_tools
'''for n < N, n = PROD(p**np) pick n such that PROD(np+1) == 1 MOD 6'''
primes = euler_tools.sieve(int(1e6))
def sieve(n):
a = [1] * n
for m in range(2, n + 1):
if m % 1e6 == 0:
print(m)
k = 1
while True:
if m * k <= len(a):
if a[m * k - 1] == 6:
a[m * k - 1] = 1
else:
a[m * k - 1] += 1
k += 1
else:
break
return a
def pairs():
i = 1
while True:
if 2**i > 1e36:
break
import euler_tools
PRIMES = euler_tools.sieve(int(3e5))
print(PRIMES[2])
def n_min(r_max):
for n, p in enumerate(PRIMES):
r = 2 * p * (1 + n)
print(r, p, n + 1, r < p**2)
if r >= r_max:
return n + 1, p, r, n, PRIMES[n - 1], 2 * PRIMES[n - 1] * n
return None
print(n_min(1e10))