def fast_sum_divisors(n):
product = 1
for factor, power in prime_factors(n, 'dictionary').items():
term = 1
for i in range(1, power + 1):
term += factor**i
product *= term
return product - n
def phi(n, limit=None):
idx = n - 1
f = prime_factors(n, 'set')
if limit and len(f) > limit: return n // 2
for i in range(len(f)):
multiplier = [-1, 1]
c = c_prod(f, i + 1)
for x in c:
idx += (n // x - 1) * multiplier[i % 2]
return idx
def m_primes(n):
f = set(prime_factors(n))
m = []
for i in range(1, n):
for p in f:
if not i % p:
break
else:
m.append(i)
# if not set(prime_factors(i)) & f:
return m
def fracs(n):
factors = prime_factors(n)
start = ceil(n / 3)
end = n // 2
for i in range(start, end+1):
if i == 1:
yield 1
continue
if i in primes and n % i:
yield i
continue
for f in factors:
if i % f == 0:
break
else:
yield i
def phi(n):
f = prime_factors(n)
if len(f) == 1:
cache[n] = n-1
return n-1
family = product(set(f))
if family > 1 and family in cache:
cache[n] = cache[family] * n // family
return cache[n]
f = set(f)
idx = n - 1
for i in range(len(f)):
multiplier = [-1, 1]
c = c_prod(f, i+1)
for x in c:
idx += (n//x - 1) * multiplier[i%2]
cache[n] = idx
return idx
def first_tri_num_with_more_than_n_factors(n):
for tri_number in triangular(None):
if 2**len(prime_factors(tri_number)) >= n:
if len(all_factors(tri_number, include_self=True)) > n:
return tri_number
from primesandfactors import prime_factors
from generators import miller_rabin_range
from benchmark import benchmark
from numpy import array
lowest_factorial_multiple = lambda n: max( ( f(v, k) for k, v in prime_factors(n, 'dictionary').items() ) )
def f(n, b):
m = n
total = 0
while n > 0:
i, e = 0, 0
while True:
x = b**e
if i + x > n: break
i += x
e += 1
n -= i
total += x
return total
@benchmark()
def run(n):
done = array( [True] * (n+1) )
total = count = 0
for prime in miller_rabin_range(n+1):
y = min(n//prime, prime)
total += prime * (y-1)
for x in range(1, y):
done[prime*x] = False
count += 1
def run2(num):
n = 1
for p in primes2(num):
n *= p
ph = phi(n)
print(p, ph, prime_factors(ph))