結果

問題 No.1781 LCM
コンテスト
ユーザー hos.lyric
提出日時 2021-12-10 00:16:55
言語 D
(dmd 2.109.1)
結果
TLE  
実行時間 -
コード長 9,407 bytes
コンパイル時間 1,510 ms
コンパイル使用メモリ 159,928 KB
実行使用メモリ 19,052 KB
最終ジャッジ日時 2024-06-22 13:35:35
合計ジャッジ時間 8,347 ms
ジャッジサーバーID
（参考情報） 		judge4 / judge2
このコードへのチャレンジ
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ファイルパターン 結果
sample -- * 2
other AC * 21 TLE * 1 -- * 9
権限があれば一括ダウンロードができます

ソースコード

diff # 

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)<p<=floor(N/l)} f(p)
  void add(S)(T coef, S primeSum) {
    assert(N == primeSum.N, "MultiplicativeSum: N must agree");
    foreach (n; 1 .. sqrtN + 1) smallFP[n] += coef * primeSum.small[n];
    foreach (l; 1 .. sqrtN + 1) {
      large[l] += coef * (primeSum.large[l] - primeSum.small[sqrtN]);
    }
  }
  // (p, e) -> 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) {
  }
}
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