def s(n):
l = decompose_tout(n, l_8)
maxi = 0
for p, k in l:
m = f(p, k)
if m > maxi:
maxi = m
return maxi
def aux(n):
l = []
for a in range(1, n + 1):
if n % a == 0:
for o, i in decompose_tout(a):
if i < 3:
break
else:
l.append(a)
return l
def circumference(n): # give the number of x, y such that x**2+y**2 = n
l = decompose_tout(n)
B = 1
for p, a in l:
if p == 2:
continue
if p % 4 == 1:
B *= a + 1
elif a % 2 == 1:
return 0
return B * 4
def nb_ok(n):
l = decompose_tout(n, l_p)
n_l = len(l)
l2 = []
nb_div = 0
for i in range(n_l):
l2.append(list(range(l[i][1] + 1)))
for c in product(*l2):
a = 1
for i in range(n_l):
a *= l[i][0]**c[i]
if 3 * a**2 - n > 0:
nb_div += 1
return nb_div
def decompose_B(n, d, df):
ln = decompose_tout(n, lp)
for p, a in ln:
if p in d:
d[p] += a * n
else:
d[p] = a * n
if p in df:
df[p] += a
else:
df[p] = a
for p in df:
d[p] -= df[p]
return sigma(d)
def s(n):
if n in l_set:
return n
d = dict(decompose_tout(n, l))
nb_tot = sum(d.values())
m = 1
nb = 0
while nb < nb_tot:
m += 1
for p in d:
mm = m
while mm % p == 0 and d[p] > 0:
mm //= p
nb += 1
d[p] -= 1
return m
from Tools import decompose_tout, bezout, numpy_sieve
l7 = numpy_sieve(10**7)
a = 1
mod = 1000000087
ss = 0
d = {}
for n in range(2, 10**7 + 1):
lp = decompose_tout(n, l7)
for p, i in lp:
if p in d:
a *= bezout(mod, 1 + 2 * d[p])[0]
d[p] += i
a *= 1 + 2 * d[p]
else:
d[p] = i
a *= 1 + 2 * d[p]
a %= mod
ss += a
ss %= mod
print(ss)
# sol = 416146418
def div(n):
ll = decompose_tout(n)
for p_a in f(ll):
yield compute(p_a)
# %% Problem 245
# C(n) can not be an unit fraction if n = p^k * a with k > 1 and p prime
from Tools import is_prime, phi, decompose_tout
def unit_co_resilience(n):
return (n - 1) % (n - phi(n)) == 0
s = 0
for i in range(2, 2 * 10**11 + 1):
if not is_prime(i):
if unit_co_resilience(i):
s += i
l = [b for a, b in decompose_tout(i)]
if l.count(1) != len(l):
print(i)
k = 1
a = 3 * c - k # that is the sol.
b = 7 * c - 2 * k # gcd(b, a) = gcd(b-2a, a) = gcd(c, 3c-1)=gcd(c, c-1)=1
# sol = 428570
# %% Problem 72
d = dict()
l_erato = numpy_sieve(10**6)
for i in l_erato:
d[i] = i - 1
s = 0
for i in range(2, 10**6 + 1):
ll = decompose_tout(i, l_erato)
p = 1
for i, k in ll:
p *= (i**(k - 1)) * d[i]
s += p
# %% Problem 73
def gcd(a, b):
if a < b:
r = b % a
return gcd(a, r)
elif b == a:
return b
elif b == 0:
def nb_div(n):
l2 = decompose_tout(n, primes)
a = 1
for c in l2:
a *= c[1] + 1
return a
