#yukicoder 2798 Multiple Chain #O(N ^ {1/4}) 高速素因数分解 def fast_fact(N): import random gcd = lambda x,y: gcd(y, x % y) if y else abs(x) #Miller-Rabin Primality Test O(logN) time, wrong rate: under 1/4 per test. def check_prime(n): if n == 1 or n % 2 == 0: return True if n == 2 else False m = n - 1; s = (m & -m).bit_length() - 1; d = m // pow(2, s) #m = d * 2^s if n < 48781 * 97561: test_number = [2, 7, 61] else: test_number = [2, 325, 9375, 28178, 450775, 9780504, 1795265022] if n > pow(2, 64): test_number.extend([random.randint(3, m) for _ in range(10)]) for a in test_number: x, r = pow(a, d, n), 0 if a == n or x == 1: continue while x != m: x, r = pow(x, 2, n), r + 1 if x == 1 or r == s: return False return True #Pollards rho algorithm def find_prime(n): if n % 2 == 0: return 2 m = int(n ** 0.125) + 1 for c in range(1, n): f, x, y, k, g, q, r = lambda x: (pow(x, 2, n) + c) % n, 0, 0, 0, 1, 1, 1 while g == 1: #y = f^k(c), x = f^{r/2}(c). especially, r is power of 2 while k < 3 * r // 4: y, k = f(y), k + 1 #skip calculate while k < r and g == 1: s = y for _ in range(min(m, r - k)): y = f(y); q = q * abs(x - y) % n g, k = gcd(q, n), k + m k, r, x = r, r * 2, y if g == n: #backtrack g, y = 1, s while g == 1: y = f(y); g = gcd(abs(x - y), n) if g == n: continue return g if check_prime(g) else n//g if check_prime(n//g) else find_prime(g) return n #RE prevention A = [] while not check_prime(N) and N > 1: P, E = find_prime(N), 0 while N % P == 0: N, E = N // P, E + 1 A.append((P, E)) if N > 1: A.append((N, 1)) return sorted(A) #入力受取 N = int(input()) #素因数分解 因数の個数を列挙 P = fast_fact(N) Q = sorted(e for f, e in P) R = max(Q) #分割数を考える #DP[n][k]: 自然数nをk個の「非負整数」の和として表す場合の数 #これは蟻本式DPでO(n^2) DP[n][k] = DP[n][k - 1] + DP[n - k][k] #k個のなかに0を含むものはDP[n][k - 1]通り #k個のなかに0を含まないものはDP[n - k][k]通り DP = [[0] * 64 for _ in range(64)] for k in range(64): DP[0][k] = 1 for n in range(1, 64): for k in range(1, 64): DP[n][k] = DP[n][k - 1] if n >= k: DP[n][k] += DP[n - k][k] #長さxの数列であって、分割を満たすものの個数。ただし、全要素の先頭が0ではだめ。 def solve(x): #1. 全要素を長さxに分割する場合の数の積を求める ans = 1 for q in Q: ans *= DP[q][x] #2. 全要素の先頭が0であるものの個数を求める if x == 1: return ans cnt = 1 for q in Q: cnt *= DP[q][x - 1] return ans - cnt ans = 0 for x in range(1, R + 1): ans += solve(x) print(ans)