import std.conv, std.functional, std.range, std.stdio, std.string; import std.algorithm, std.array, std.bigint, std.bitmanip, std.complex, std.container, std.math, std.mathspecial, std.numeric, std.regex, std.typecons; import core.bitop; class EOFException : Throwable { this() { super("EOF"); } } string[] tokens; string readToken() { for (; tokens.empty; ) { if (stdin.eof) { throw new EOFException; } tokens = readln.split; } auto token = tokens.front; tokens.popFront; return token; } int readInt() { return readToken.to!int; } long readLong() { return readToken.to!long; } real readReal() { return readToken.to!real; } bool chmin(T)(ref T t, in T f) { if (t > f) { t = f; return true; } else { return false; } } bool chmax(T)(ref T t, in T f) { if (t < f) { t = f; return true; } else { return false; } } int binarySearch(alias pred, T)(in T[] as) { int lo = -1, hi = cast(int)(as.length); for (; lo + 1 < hi; ) { const mid = (lo + hi) >> 1; (unaryFun!pred(as[mid]) ? hi : lo) = mid; } return hi; } int lowerBound(T)(in T[] as, T val) { return as.binarySearch!(a => (a >= val)); } int upperBound(T)(in T[] as, T val) { return as.binarySearch!(a => (a > val)); } struct ModInt(uint M_) { import std.conv : to; alias M = M_; uint x; this(ModInt a) { x = a.x; } this(uint x_) { x = x_ % M; } this(ulong x_) { x = x_ % M; } this(int x_) { x = ((x_ %= cast(int)(M)) < 0) ? (x_ + cast(int)(M)) : x_; } this(long x_) { x = cast(uint)(((x_ %= cast(long)(M)) < 0) ? (x_ + cast(long)(M)) : x_); } ref ModInt opAssign(T)(inout(T) a) if (is(T == uint) || is(T == ulong) || is(T == int) || is(T == long)) { return this = ModInt(a); } ref ModInt opOpAssign(string op, T)(T a) { static if (is(T == ModInt)) { static if (op == "+") { x = ((x += a.x) >= M) ? (x - M) : x; } else static if (op == "-") { x = ((x -= a.x) >= M) ? (x + M) : x; } else static if (op == "*") { x = cast(uint)((cast(ulong)(x) * a.x) % M); } else static if (op == "/") { this *= a.inv(); } else static assert(false); return this; } else static if (op == "^^") { if (a < 0) return this = inv()^^(-a); ModInt b = this, c = 1U; for (long e = a; e; e >>= 1) { if (e & 1) c *= b; b *= b; } return this = c; } else { return mixin("this " ~ op ~ "= ModInt(a)"); } } ModInt inv() const { uint a = M, b = x; int y = 0, z = 1; for (; b; ) { const q = a / b; const c = a - q * b; a = b; b = c; const w = y - cast(int)(q) * z; y = z; z = w; } assert(a == 1); return ModInt(y); } ModInt opUnary(string op)() const { static if (op == "+") { return this; } else static if (op == "-") { ModInt a; a.x = x ? (M - x) : 0U; return a; } else static assert(false); } ModInt opBinary(string op, T)(T a) const { return mixin("ModInt(this) " ~ op ~ "= a"); } ModInt opBinaryRight(string op, T)(T a) const { return mixin("ModInt(a) " ~ op ~ "= this"); } bool opCast(T: bool)() const { return (x != 0U); } string toString() const { return x.to!string; } } enum MO = 998244353; alias Mint = ModInt!MO; // floor(sqrt(a)) long floorSqrt(long a) { import core.bitop : bsr; import std.algorithm : min; long b = a, x = 0, y = 0; for (int e = bsr(a) & ~1; e >= 0; e -= 2) { x <<= 1; y <<= 1; if (b >= (y | 1) << e) { b -= (y | 1) << e; x |= 1; y += 2; } } return x; } // get(floor(N / l)) = \sum_{p<=floor(N/l)} p^K // O(N^(3/4) / log N) time, O(N^(1/2)) space class PrimeSum(T, int K) { long N, sqrtN; bool[] isPrime; T[] small, large; this(long N) { assert(N >= 1, "PrimeSum: N >= 1 must hold"); this.N = N; sqrtN = floorSqrt(N); isPrime = new bool[sqrtN + 1]; small = new T[sqrtN + 1]; large = new T[sqrtN + 1]; isPrime[2 .. $] = true; T powerSum(long n) { static if (K == 0) { return T(n); } else static if (K == 1) { long n0 = n, n1 = n + 1; ((n0 % 2 == 0) ? n0 : n1) /= 2; return T(n0) * T(n1); } else static if (K == 2) { long n0 = n, n1 = n + 1, n2 = 2 * n + 1; ((n0 % 2 == 0) ? n0 : n1) /= 2; ((n0 % 3 == 0) ? n0 : (n1 % 3 == 0) ? n1 : n2) /= 3; return T(n0) * T(n1) * T(n2); } else static if (K == 3) { long n0 = n, n1 = n + 1; ((n0 % 2 == 0) ? n0 : n1) /= 2; return T(n0) * T(n0) * T(n1) * T(n1); } else { static assert(false, "PrimeSum: K is out of range"); } } foreach (n; 1 .. sqrtN + 1) small[n] = powerSum(n); foreach (l; 1 .. sqrtN + 1) large[l] = powerSum(N / l); foreach (p; 2 .. sqrtN + 1) { if (isPrime[p]) { for (long n = p^^2; n <= sqrtN; n += p) isPrime[n] = false; const pk = T(p)^^K, g1 = get(p - 1); foreach (l; 1 .. sqrtN + 1) { const n = N / l; if (n < p^^2) break; large[l] -= pk * (get(n / p) - g1); } foreach_reverse (n; 1 .. sqrtN + 1) { if (n < p^^2) break; small[n] -= pk * (get(n / p) - g1); } } } small[1 .. $] -= T(1); large[1 .. $] -= T(1); } T get(long n) const { return (n <= sqrtN) ? small[n] : large[N / n]; } } // get(floor(N / l)) = \sum_{p<=floor(N/l)} p^K // O(N^(3/4) / log N) time, O(N^(1/2)) space // large K; \sum_{i=1}^n i^K = \sum_{j=1}^{K+1} coef[j] n^j class PrimeSum(T) { long N, sqrtN; bool[] isPrime; T[] small, large; this(long N, int K, T[] coef) { assert(N >= 1, "PrimeSum: N >= 1 must hold"); this.N = N; sqrtN = floorSqrt(N); isPrime = new bool[sqrtN + 1]; small = new T[sqrtN + 1]; large = new T[sqrtN + 1]; isPrime[2 .. $] = true; T powerSum(long n) { T y = 0; foreach_reverse (k; 1 .. K + 2) (y += coef[k]) *= n; return y; } foreach (n; 1 .. sqrtN + 1) small[n] = powerSum(n); foreach (l; 1 .. sqrtN + 1) large[l] = powerSum(N / l); foreach (p; 2 .. sqrtN + 1) { if (isPrime[p]) { for (long n = p^^2; n <= sqrtN; n += p) isPrime[n] = false; const pk = T(p)^^K, g1 = get(p - 1); foreach (l; 1 .. sqrtN + 1) { const n = N / l; if (n < p^^2) break; large[l] -= pk * (get(n / p) - g1); } foreach_reverse (n; 1 .. sqrtN + 1) { if (n < p^^2) break; small[n] -= pk * (get(n / p) - g1); } } } small[1 .. $] -= T(1); large[1 .. $] -= T(1); } T get(long n) const { return (n <= sqrtN) ? small[n] : large[N / n]; } } // get(floor(N / l)) = \sum_{n=1}^{floor(N/l)} f(n) // O(N^(3/4) / log N) time, O(N^(1/2)) space // f: multiplicative function, f(p): poly in p class MultiplicativeSum(T) { long N, sqrtN; bool[] isPrime; T[] smallFP, small, large; this(long N) { assert(N >= 1, "PrimeSum: N >= 1 must hold"); this.N = N; sqrtN = floorSqrt(N); isPrime = new bool[sqrtN + 1]; smallFP = new T[sqrtN + 1]; small = new T[sqrtN + 1]; large = new T[sqrtN + 1]; isPrime[2 .. $] = true; foreach (p; 2 .. sqrtN + 1) { if (isPrime[p]) { for (long n = p^^2; n <= sqrtN; n += p) isPrime[n] = false; } } } // prepare \sum_{p<=n} f(p) and \sum_{N^(1/2) f(p^e) void build(T delegate(long, int) f) { import std.algorithm : max; small[1 .. $] += T(1); large[1 .. $] += T(1); foreach_reverse (p; 2 .. sqrtN + 1) { if (isPrime[p]) { // added f(p') for p < p' <= min{n, N^(1/2)} T getAdded(long n) const { return (n <= sqrtN) ? (small[n] + smallFP[max(n, p)] - smallFP[p]) : (large[N / n] + smallFP[sqrtN] - smallFP[p]); } // p^e, f(p^e) long[] pes = [1]; T[] fs = [T(1)]; long pe = p; for (int e = 1; ; ++e) { pes ~= pe; fs ~= f(p, e); if (pe > N / p) break; pe *= p; } const limE = cast(int)(pes.length); foreach (l; 1 .. sqrtN + 1) { const n = N / l; if (n < p^^2) break; for (int e = 1; e < limE && pes[e] <= n; ++e) { large[l] += fs[e] * getAdded(n / pes[e]); } large[l] -= fs[1]; } foreach_reverse (n; 1 .. sqrtN + 1) { if (n < p^^2) break; for (int e = 1; e < limE && pes[e] <= n; ++e) { small[n] += fs[e] * getAdded(n / pes[e]); } small[n] -= fs[1]; } } } small[] += smallFP[]; large[1 .. $] += smallFP[sqrtN]; } T get(long n) const { return (n <= sqrtN) ? small[n] : large[N / n]; } } enum E = 40; void main() { try { for (; ; ) { const N = readLong(); const M = readLong(); auto ms = new MultiplicativeSum!Mint(M); ms.add(Mint(1 + 1)^^N, new PrimeSum!(Mint, 0)(M)); auto pw = new Mint[E]; foreach (e; 0 .. E) { pw[e] = Mint(e + 1)^^N; } ms.build((p, e) => pw[e]); const Mint ans = ms.get(M); writeln(ans); } } catch (EOFException e) { } }