#pragma GCC target("avx2") #pragma GCC optimize("O3") #pragma GCC optimize("unroll-loops") #include #define N_MAX 200000 #define M_MAX 200000 const int Mod = 998244353, bit[21] = {1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576}, bit_inv[21] = {1, 499122177, 748683265, 873463809, 935854081, 967049217, 982646785, 990445569, 994344961, 996294657, 997269505, 997756929, 998000641, 998122497, 998183425, 998213889, 998229121, 998236737, 998240545, 998242449, 998243401}, root[21] = {1, 998244352, 911660635, 372528824, 929031873, 452798380, 922799308, 781712469, 476477967, 166035806, 258648936, 584193783, 63912897, 350007156, 666702199, 968855178, 629671588, 24514907, 996173970, 363395222, 565042129}, root_inv[21] = {1, 998244352, 86583718, 509520358, 337190230, 87557064, 609441965, 135236158, 304459705, 685443576, 381598368, 335559352, 129292727, 358024708, 814576206, 708402881, 283043518, 3707709, 121392023, 704923114, 950391366}; int ntt_b[20][524288], ntt_c[20][524288], ntt_x[20][524288], ntt_y[20][524288]; void NTT(int k, int a[], int z[]) { if (k == 0) { z[0] = a[0]; return; } int i, d = bit[k-1], tmpp; long long tmp; for (i = 0; i < d; i++) { ntt_b[k][i] = a[i*2]; ntt_c[k][i] = a[i*2+1]; } NTT(k - 1, ntt_b[k], ntt_x[k]); NTT(k - 1, ntt_c[k], ntt_y[k]); for (i = 0, tmp = 1; i < d; i++, tmp = tmp * root[k] % Mod) { tmpp = tmp * ntt_y[k][i] % Mod; z[i] = ntt_x[k][i] + tmpp; if (z[i] >= Mod) z[i] -= Mod; z[i+d] = ntt_x[k][i] - tmpp; if (z[i+d] < 0) z[i+d] += Mod; } } void NTT_reverse(int k, int z[], int a[]) { if (k == 0) { a[0] = z[0]; return; } int i, d = bit[k-1], tmpp; long long tmp; for (i = 0; i < d; i++) { ntt_x[k][i] = z[i*2]; ntt_y[k][i] = z[i*2+1]; } NTT_reverse(k - 1, ntt_x[k], ntt_b[k]); NTT_reverse(k - 1, ntt_y[k], ntt_c[k]); for (i = 0, tmp = 1; i < d; i++, tmp = tmp * root_inv[k] % Mod) { tmpp = tmp * ntt_c[k][i] % Mod; a[i] = ntt_b[k][i] + tmpp; if (a[i] >= Mod) a[i] -= Mod; a[i+d] = ntt_b[k][i] - tmpp; if (a[i+d] < 0) a[i+d] += Mod; } } // Compute the product of two polynomials a[0-da] and b[0-db] using NTT in O(d * log d) time void prod_poly_NTT(int da, int db, int a[], int b[], int c[]) { int i, k; for (k = 0; bit[k] <= da + db; k++); for (i = da + 1; i < bit[k]; i++) a[i] = 0; for (i = db + 1; i < bit[k]; i++) b[i] = 0; static int x[524288], y[524288], z[524288]; NTT(k, a, x); if (db == da) { for (i = 0; i <= da; i++) if (a[i] != b[i]) break; if (i <= da) NTT(k, b, y); else for (i = 0; i < bit[k]; i++) y[i] = x[i]; } else NTT(k, b, y); for (i = 0; i < bit[k]; i++) z[i] = (long long)x[i] * y[i] % Mod; NTT_reverse(k, z, c); for (i = 0; i <= da + db; i++) c[i] = (long long)c[i] * bit_inv[k] % Mod; } // Compute the product of two polynomials a[0-da] and b[0-db] naively in O(da * db) time void prod_poly_naive(int da, int db, int a[], int b[], int c[]) { int i, j; for (i = 0; i <= da + db; i++) c[i] = 0; for (i = 0; i <= da; i++) { for (j = 0; j <= db; j++) { c[i+j] += (long long)a[i] * b[j] % Mod; if (c[i+j] >= Mod) c[i+j] -= Mod; } } } // Compute the product of two polynomials a[0-da] and b[0-db] in an appropriate way void prod_polynomial(int da, int db, int a[], int b[], int c[]) { const int THR = 250000; if (THR / (da + 1) >= db + 1) prod_poly_naive(da, db, a, b, c); else prod_poly_NTT(da, db, a, b, c); } long long pow_mod(int n, long long k) { long long N, ans = 1; for (N = n; k > 0; k >>= 1, N = N * N % Mod) if (k & 1) ans = ans * N % Mod; return ans; } int solve_sqrt(int N, int M) { const int THR = 10000; int i, j, d[N_MAX + 1] = {}, a[524288], b[524288], c[524288] = {}; long long dp[N_MAX + 1], tmp, sum = 0; for (i = 2; i <= M; i++) for (j = i; j <= N; j += i) d[j]++; for (j = 0; j < THR; j++) b[j] = d[j]; for (i = 1, dp[0] = 1; i <= N; i++) { if (i % THR == 0) { for (j = (i - 1) / THR * THR; j < i; j++) { a[j] = dp[j]; b[j+THR] = d[j+THR]; } prod_poly_NTT(i - 1, i - 1 + THR, a, b, c); } for (j = i / THR * THR, tmp = c[i]; j < i - 1; j++) tmp += dp[j] * d[i-j]; tmp %= Mod; dp[i] = pow_mod(M - 1, i) - (sum + tmp) % Mod; if (dp[i] < 0) dp[i] += Mod; sum = (sum * (M - 1) + tmp * (M - 2)) % Mod; } return (pow_mod(M, N) - dp[N] % Mod + Mod) % Mod; } long long recursion(int N, int M, int d[], long long dp[], long long tmp[], long long sum, int l, int r) { if (l == r) { tmp[l] %= Mod; dp[l] = pow_mod(M - 1, l) - (sum + tmp[l]) % Mod; if (dp[l] < 0) dp[l] += Mod; sum = (sum * (M - 1) + tmp[l] * (M - 2)) % Mod; return sum; } int i, m = (l + r) / 2; static int a[524288], b[524288], c[524288]; sum = recursion(N, M, d, dp, tmp, sum, l, m); for (i = l; i <= m; i++) a[i-l] = dp[i]; for (i = l; i <= r; i++) b[i-l] = d[i-l]; prod_polynomial(m - l, r - l, a, b, c); for (i = m + 1; i <= r; i++) tmp[i] += c[i-l]; return recursion(N, M, d, dp, tmp, sum, m + 1, r); } int solve_log2(int N, int M) { int i, j, d[N_MAX + 1] = {}; long long dp[N_MAX + 1] = {1}, tmp[N_MAX + 1] = {}; for (i = 2; i <= M; i++) for (j = i; j <= N; j += i) d[j]++; for (i = 1; i <= N; i++) tmp[i] = d[i]; recursion(N, M, d, dp, tmp, 0, 1, N); return (pow_mod(M, N) - dp[N] % Mod + Mod) % Mod; } int main() { int N, M; scanf("%d %d", &N, &M); // printf("%d\n", solve_sqrt(N, M)); printf("%d\n", solve_log2(N, M)); fflush(stdout); return 0; }