def test_core():
assert core(35**13, 10) == 42875
assert core(210**2) == 1
assert core(7776, 3) == 36
assert core(10**27, 22) == 10**5
assert core(537824) == 14
assert core(1, 6) == 1
def test_core():
assert core(35**13, 10) == 42875
assert core(210**2) == 1
assert core(7776, 3) == 36
assert core(10**27, 22) == 10**5
assert core(537824) == 14
assert core(1, 6) == 1
def a007913(n: int) -> int:
"""
https://oeis.org/A007913
>>> a007913(1)
1
>>> a007913(8)
2
>>> a007913(24)
6
"""
from sympy.ntheory.factor_ import core
return core(n)
def _regular_point_ellipse(self, a, b, c, d, e, f):
D = 4 * a * c - b**2
ok = D
if not ok:
raise ValueError("Rational Point on the conic does not exist")
if a == 0 and c == 0:
K = -1
L = 4 * (d * e - b * f)
elif c != 0:
K = D
L = 4 * c**2 * d**2 - 4 * b * c * d * e + 4 * a * c * e**2 + 4 * b**2 * c * f - 16 * a * c**2 * f
else:
K = D
L = 4 * a**2 * e**2 - 4 * b * a * d * e + 4 * b**2 * a * f
ok = L != 0 and not (K > 0 and L < 0)
if not ok:
raise ValueError("Rational Point on the conic does not exist")
K = Rational(K).limit_denominator(10**12)
L = Rational(L).limit_denominator(10**12)
k1, k2 = K.p, K.q
l1, l2 = L.p, L.q
g = gcd(k2, l2)
a1 = (l2 * k2) / g
b1 = (k1 * l2) / g
c1 = -(l1 * k2) / g
a2 = sign(a1) * core(abs(a1), 2)
r1 = sqrt(a1 / a2)
b2 = sign(b1) * core(abs(b1), 2)
r2 = sqrt(b1 / b2)
c2 = sign(c1) * core(abs(c1), 2)
r3 = sqrt(c1 / c2)
g = gcd(gcd(a2, b2), c2)
a2 = a2 / g
b2 = b2 / g
c2 = c2 / g
g1 = gcd(a2, b2)
a2 = a2 / g1
b2 = b2 / g1
c2 = c2 * g1
g2 = gcd(a2, c2)
a2 = a2 / g2
b2 = b2 * g2
c2 = c2 / g2
g3 = gcd(b2, c2)
a2 = a2 * g3
b2 = b2 / g3
c2 = c2 / g3
x, y, z = symbols("x y z")
eq = a2 * x**2 + b2 * y**2 + c2 * z**2
solutions = diophantine(eq)
if len(solutions) == 0:
raise ValueError("Rational Point on the conic does not exist")
flag = False
for sol in solutions:
syms = Tuple(*sol).free_symbols
rep = {s: 3 for s in syms}
sol_z = sol[2]
if sol_z == 0:
flag = True
continue
if not (isinstance(sol_z, Integer) or isinstance(sol_z, int)):
syms_z = sol_z.free_symbols
if len(syms_z) == 1:
p = next(iter(syms_z))
p_values = Complement(
S.Integers, solveset(Eq(sol_z, 0), p, S.Integers))
rep[p] = next(iter(p_values))
if len(syms_z) == 2:
p, q = list(ordered(syms_z))
for i in S.Integers:
subs_sol_z = sol_z.subs(p, i)
q_values = Complement(
S.Integers,
solveset(Eq(subs_sol_z, 0), q, S.Integers))
if not q_values.is_empty:
rep[p] = i
rep[q] = next(iter(q_values))
break
if len(syms) != 0:
x, y, z = tuple(s.subs(rep) for s in sol)
else:
x, y, z = sol
flag = False
break
if flag:
raise ValueError("Rational Point on the conic does not exist")
x = (x * g3) / r1
y = (y * g2) / r2
z = (z * g1) / r3
x = x / z
y = y / z
if a == 0 and c == 0:
x_reg = (x + y - 2 * e) / (2 * b)
y_reg = (x - y - 2 * d) / (2 * b)
elif c != 0:
x_reg = (x - 2 * d * c + b * e) / K
y_reg = (y - b * x_reg - e) / (2 * c)
else:
y_reg = (x - 2 * e * a + b * d) / K
x_reg = (y - b * y_reg - d) / (2 * a)
return x_reg, y_reg
