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