from typing import List, Dict, Tuple def setMebius(n: int) -> List[int]: is_prime: List[int] = [True] * (n + 1) mebius: List[int] = [1] * (n + 1) is_prime[0] = False is_prime[1] = False for i in range(2, n + 1): if is_prime[i]: bai: int = 0 if i == 2 else 2 mebius[i] *= -1 for j in range(2 * i, n + 1, i): is_prime[j] = False if bai == 0: mebius[j] = 0 else: mebius[j] *= -1 bai += 1 if bai == i: bai = 0 return mebius def calc(L: int, noDiv2: bool, noDiv3: bool, mebius: List[int]) -> int: ret: int = 0 i: int = 1 while i * i <= L: keisu: int = 0 # g(i) := L以下の正のi * iの倍数であって「noDiv*=trueなら*の倍数でない」を満たすものの個数 if i % 2 == 0 and noDiv2: keisu = 0 elif i % 3 == 0 and noDiv3: keisu = 0 else: c: int = L // (i * i) keisu = c if noDiv2: keisu -= c // 2 if noDiv3: keisu -= c // 3 if noDiv2 and noDiv3: keisu += c // 6 # f(i) := L以下の正のi * iの倍数であって「noDiv*=trueなら*の倍数でない」を満たす上で、任意の整数j > iについてj * jの倍数でもないものの個数 # f(i) = g(i) - f(2 * i) - f(3 * i) - f(4 * i) - ...より、g(i) = sum(i|j, f(j)) (jはiの倍数) # メビウスの反転公式より f(i) = sum(i|j, μ(j/i) g(j)). 求めたいのはf(1)なので、f(1) = sum(j, μ(j)g(j)) が得られる. ret += mebius[i] * keisu i += 1 return ret def subtask4(L: int, a: List[int], mebius: List[int]) -> int: n: int = len(a) plot: List[List[bool]] = [[False for _ in range(60)] for _ in range(60)] for i in range(n): c2: int = 0 c3: int = 0 t: int = a[i] while t % 2 == 0: t //= 2 c2 += 1 while t % 3 == 0: t //= 3 c3 += 1 plot[c2][c3] = True p2: List[int] = [1] p3: List[int] = [1] while p2[-1] <= L: p2.append(p2[-1] * 2) while p3[-1] <= L: p3.append(p3[-1] * 3) ret: int = 0 i: int = 0 mp: Dict[Tuple[int, int, int], int] = {} while p2[i] <= L: j: int = 0 while p2[i] * p3[j] <= L: tmpL: int = L // (p2[i] * p3[j]) # 選ぶ要素のLCM = p2[i] * p3[j] for k in range(16): max2: int = -1 min2: int = 1000000 max3: int = -1 min3: int = 1000000 cnt: int = 0 not_found: bool = False for l in range(4): if (k >> l) % 2 == 1: c2: int = i - l // 2 c3: int = j - l % 2 if c2 < 0 or c3 < 0 or plot[c2][c3] == False: not_found = True break max2 = max(max2, c2) min2 = min(min2, c2) max3 = max(max3, c3) min3 = min(min3, c3) cnt += 1 if not_found or cnt == 0 or max2 != i or max3 != j: continue noDiv2 = (min2 != max2) noDiv3 = (min3 != max3) key = (tmpL, noDiv2, noDiv3) if key not in mp: mp[key] = calc(tmpL, noDiv2, noDiv3, mebius) if cnt % 2 == 1: ret += mp[key] * p2[cnt - 1] else: ret -= mp[key] * p2[cnt - 1] j += 1 i += 1 return ret if __name__ == '__main__': L, N = map(int, input().split()) a = list(map(int, input().split())) sqL: int = 1 while sqL * sqL <= L: sqL += 1 sqL -= 1 mebius: List[int] = setMebius(sqL) ans: int = subtask4(L, a, mebius) print(ans)