結果

問題 No.2620 Sieve of Coins
ユーザー ecottea
提出日時 2024-01-27 17:50:04
言語 C++17
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 1,598 ms / 2,000 ms
コード長 15,232 bytes
コンパイル時間 5,854 ms
コンパイル使用メモリ 301,700 KB
最終ジャッジ日時 2025-02-19 00:27:59
ジャッジサーバーID
(参考情報)
judge2 / judge2
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ファイルパターン 結果
sample AC * 5
other AC * 53
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ソースコード

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#ifndef HIDDEN_IN_VS //
//
#define _CRT_SECURE_NO_WARNINGS
//
#include <bits/stdc++.h>
using namespace std;
//
using ll = long long; using ull = unsigned long long; // -2^63 2^63 = 9 * 10^18int -2^31 2^31 = 2 * 10^9
using pii = pair<int, int>; using pll = pair<ll, ll>; using pil = pair<int, ll>; using pli = pair<ll, int>;
using vi = vector<int>; using vvi = vector<vi>; using vvvi = vector<vvi>; using vvvvi = vector<vvvi>;
using vl = vector<ll>; using vvl = vector<vl>; using vvvl = vector<vvl>; using vvvvl = vector<vvvl>;
using vb = vector<bool>; using vvb = vector<vb>; using vvvb = vector<vvb>;
using vc = vector<char>; using vvc = vector<vc>; using vvvc = vector<vvc>;
using vd = vector<double>; using vvd = vector<vd>; using vvvd = vector<vvd>;
template <class T> using priority_queue_rev = priority_queue<T, vector<T>, greater<T>>;
using Graph = vvi;
//
const double PI = acos(-1);
const vi DX = { 1, 0, -1, 0 }; // 4
const vi DY = { 0, 1, 0, -1 };
int INF = 1001001001; ll INFL = 4004004003104004004LL; // (int)INFL = 1010931620;
//
struct fast_io { fast_io() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(18); } } fastIOtmp;
//
#define all(a) (a).begin(), (a).end()
#define sz(x) ((int)(x).size())
#define lbpos(a, x) (int)distance((a).begin(), std::lower_bound(all(a), x))
#define ubpos(a, x) (int)distance((a).begin(), std::upper_bound(all(a), x))
#define Yes(b) {cout << ((b) ? "Yes\n" : "No\n");}
#define YES(b) {cout << ((b) ? "YES\n" : "NO\n");}
#define rep(i, n) for(int i = 0, i##_len = int(n); i < i##_len; ++i) // 0 n-1
#define repi(i, s, t) for(int i = int(s), i##_end = int(t); i <= i##_end; ++i) // s t
#define repir(i, s, t) for(int i = int(s), i##_end = int(t); i >= i##_end; --i) // s t
#define repe(v, a) for(const auto& v : (a)) // a
#define repea(v, a) for(auto& v : (a)) // a
#define repb(set, d) for(int set = 0, set##_ub = 1 << int(d); set < set##_ub; ++set) // d
#define repis(i, set) for(int i = lsb(set), bset##i = set; i >= 0; bset##i -= 1 << i, i = lsb(bset##i)) // set
#define repp(a) sort(all(a)); for(bool a##_perm = true; a##_perm; a##_perm = next_permutation(all(a))) // a
#define smod(n, m) ((((n) % (m)) + (m)) % (m)) // mod
#define uniq(a) {sort(all(a)); (a).erase(unique(all(a)), (a).end());} //
#define EXIT(a) {cout << (a) << endl; exit(0);} //
#define inQ(x, y, u, l, d, r) ((u) <= (x) && (l) <= (y) && (x) < (d) && (y) < (r)) //
//
template <class T> inline ll pow(T n, int k) { ll v = 1; rep(i, k) v *= n; return v; }
template <class T> inline bool chmax(T& M, const T& x) { if (M < x) { M = x; return true; } return false; } // true
    
template <class T> inline bool chmin(T& m, const T& x) { if (m > x) { m = x; return true; } return false; } // true
    
template <class T> inline T get(T set, int i) { return (set >> i) & T(1); }
//
template <class T, class U> inline istream& operator>>(istream& is, pair<T, U>& p) { is >> p.first >> p.second; return is; }
template <class T> inline istream& operator>>(istream& is, vector<T>& v) { repea(x, v) is >> x; return is; }
template <class T> inline vector<T>& operator--(vector<T>& v) { repea(x, v) --x; return v; }
template <class T> inline vector<T>& operator++(vector<T>& v) { repea(x, v) ++x; return v; }
#endif //
#if __has_include(<atcoder/all>)
#include <atcoder/all>
using namespace atcoder;
#ifdef _MSC_VER
#include "localACL.hpp"
#endif
//using mint = modint1000000007;
using mint = modint998244353;
//using mint = modint; // mint::set_mod(m);
namespace atcoder {
inline istream& operator>>(istream& is, mint& x) { ll x_; is >> x_; x = x_; return is; }
inline ostream& operator<<(ostream& os, const mint& x) { os << x.val(); return os; }
}
using vm = vector<mint>; using vvm = vector<vm>; using vvvm = vector<vvm>; using vvvvm = vector<vvvm>;
#endif
#ifdef _MSC_VER // Visual Studio
#include "local.hpp"
#else // gcc
inline int popcount(int n) { return __builtin_popcount(n); }
inline int popcount(ll n) { return __builtin_popcountll(n); }
inline int lsb(int n) { return n != 0 ? __builtin_ctz(n) : -1; }
inline int lsb(ll n) { return n != 0 ? __builtin_ctzll(n) : -1; }
inline int msb(int n) { return n != 0 ? (31 - __builtin_clz(n)) : -1; }
inline int msb(ll n) { return n != 0 ? (63 - __builtin_clzll(n)) : -1; }
#define dump(...)
#define dumpel(v)
#define dump_list(v)
#define dump_mat(v)
#define input_from_file(f)
#define output_to_file(f)
#define Assert(b) { if (!(b)) while (1) cout << "OLE"; }
#endif
ll solve_small_L(ll l, int n, vl a) {
vi c(l + 1);
repe(x, a) c[x] = 1;
repi(i, 1, l) {
if (!c[i]) continue;
for (int x = i * 2; x <= l; x += i) {
c[x] ^= 1;
}
}
dump(c);
return accumulate(all(c), 0LL);
}
//O(√n)
/*
* μ(n)
* μ(n) = (-1)^kn k or 0n
*/
int mobius_mu(ll n) {
// verify : https://algo-method.com/tasks/494
int res = 1;
for (ll i = 2; i * i <= n; i++) {
if (n % (i * i) == 0) return 0;
if (n % i == 0) {
n /= i;
res *= -1;
}
}
if (n > 1) res *= -1;
return res;
}
void zikken() {
int n = 1000;
vi c(n + 1);
c[1] = 1;
repi(i, 1, n) {
if (!c[i]) continue;
for (int x = i * 2; x <= n; x += i) {
c[x] ^= 1;
}
}
repi(i, 1, n) {
// cout << c[i] << " " << mobius_mu(i) << endl;
assert(c[i] == abs(mobius_mu(i)));
}
exit(0);
}
//
/*
* Multiplicative_dirichlet_convolution_acc<T>(int p_max) : O(p_max log(log p_max))
* p_max ≧ nl
* a[1..n] b[1..n] c[1..n]
* nl, nh nh ≦ nl ≦ n ≦ nl nh al, bl, cl, Ah, Bh, Ch
* al[i] = a[i] (i∈[1..nl]), bl, cl
* Ah[i] = Σa[1..n/i] (i∈[1..nh]), Bh, Ch
*
* conv_acc(ll n, vT al, vT Ah, vT bl, vT Bh, vT& cl, vT& Ch) : O(nl log(log nl) + √(n nh))
* al, Ah, bl, Bh cl, Ch
*
* inv_conv_acc(ll n, vT al, vT Ah, vT cl, vT Ch, vT& bl, vT& Bh) : O(nl log(log nl) + √(n nh))
* al, Ah, cl, Ch bl, Bh
*
* nl = (n / log(log n))^(2/3) O(n^(2/3) (log(log n))^(1/3))
int n_max = (int)1e8;
int nl = min((int)pow(n_max / log(log(n_max)), 2. / 3), (int)n);
int nh = min((int)n / nl + 1, nl);
*/
template <class T>
class Multiplicative_dirichlet_convolution_acc {
// : https://maspypy.com/dirichlet-%e7%a9%8d%e3%81%a8%e3%80%81%e6%95%b0%e8%ab%96%e9%96%a2%e6%95%b0%e3%81%ae%e7%b4%af%e7%a9%8d%e5%92%8c
int p_max;
vi ps; //
public:
// nl
Multiplicative_dirichlet_convolution_acc(int p_max) : p_max(p_max) {
// verify : https://judge.yosupo.jp/problem/sum_of_totient_function
// is_prime[i] : i
vb is_prime(p_max + 1, true);
is_prime[0] = is_prime[1] = false;
int i = 2;
// √p_max i
for (; i <= p_max / i; i++) if (is_prime[i]) {
ps.push_back(i);
for (int j = i * i; j <= p_max; j += i) is_prime[j] = false;
}
// √p_max i
for (; i <= p_max; i++) if (is_prime[i]) ps.push_back(i);
}
// al, Ah, bl, Bh cl, Ch
void conv_acc(ll n, const vector<T>& al, const vector<T>& Ah,
const vector<T>& bl, const vector<T>& Bh, vector<T>& cl, vector<T>& Ch)
{
int nl = sz(al) - 1, nh = sz(Ah) - 1;
Assert(nl <= p_max); Assert(nh <= nl); Assert(nl <= n); Assert(n <= (ll)nl * nh);
cl = bl, Ch.assign(nh + 1, 0);
// cl[1..nl]
repe(p, ps) repir(j, nl / p, 1) {
for (ll i = p; i * j <= nl; i *= p) cl[i * j] += al[i] * cl[j];
}
// Al[i] = Σa[1..i], Bl[i] = Σb[1..i]
vector<T> Al(nl + 1), Bl(nl + 1);
repi(i, 1, nl) {
Al[i] = Al[i - 1] + al[i];
Bl[i] = Bl[i - 1] + bl[i];
}
auto get_Ah = [&](ll i) { return i <= nh ? Ah[i] : Al[n / i]; };
auto get_Bh = [&](ll i) { return i <= nh ? Bh[i] : Bl[n / i]; };
// Ch[k]
repi(k, 1, nh) {
int m = (int)(sqrt(n / k) + 1e-12);
repi(i, 1, m) Ch[k] += al[i] * get_Bh((ll)k * i);
repi(j, 1, m) Ch[k] += bl[j] * (get_Ah((ll)k * j) - Al[m]);
}
}
// al, Ah, cl, Ch bl, Bh
void inv_conv_acc(ll n, const vector<T>& al, const vector<T>& Ah,
vector<T>& bl, vector<T>& Bh)
{
// verify : https://judge.yosupo.jp/problem/sum_of_totient_function
Assert(al[1] != 0);
int nl = sz(al) - 1, nh = sz(Ah) - 1;
Assert(nl <= p_max); Assert(nh <= nl); Assert(nl <= n); Assert(n <= (ll)nl * nh);
// bl[1..nl]
repe(p, ps) repi(j, 1, nl / p) {
bl[j] /= al[1];
for (ll i = p; i * j <= nl; i *= p) bl[i * j] -= al[i] * bl[j];
}
// Al[i] = Σa[1..i], Bl[i] = Σb[1..i]
vector<T> Al(nl + 1), Bl(nl + 1);
repi(i, 1, nl) {
Al[i] = Al[i - 1] + al[i];
Bl[i] = Bl[i - 1] + bl[i];
}
auto get_Ah = [&](ll i) { return i <= nh ? Ah[i] : Al[n / i]; };
auto get_Bh = [&](ll i) { return i <= nh ? Bh[i] : Bl[n / i]; };
// Bh[k]
repir(k, nh, 1) {
int m = (int)(sqrt(n / k) + 1e-12);
repi(i, 2, m) Bh[k] -= al[i] * get_Bh((ll)k * i);
repi(j, 1, m) Bh[k] -= bl[j] * (get_Ah((ll)k * j) - Al[m]);
Bh[k] /= al[1];
}
}
};
//O(n^(2/3) log(log n)^(1/3))
/*
* i∈[1..nl] bl[i] = μ(i)^2
* i∈[1..nh] Bh[i] = Σj∈[1..n/i] μ(j)^2
*
* nh ≦ nl ≦ n ≦ nl nh
*
*
*/
ll square_free_sum(ll n, int nl, int nh) {
// : https://maspypy.com/dirichlet-%e7%a9%8d%e3%81%a8%e3%80%81%e6%95%b0%e8%ab%96%e9%96%a2%e6%95%b0%e3%81%ae%e7%b4%af%e7%a9%8d%e5%92%8c
//
// μ(i)^2 M(s)
// M(s) = Π_p (1 + 1/p^s)
// = Π_p (1 - 1/p^(2s)) / Π_p (1 - 1/p^s)
// = Π_p 1/(1 - 1/p^s) / Π_p 1/(1 - 1/p^(2s))
// = ζ(s) / ζ(2s)
//
// ζ(2s) a[i] = (i ? 1 : 0)
// ζ(s) c[i] = 1
if (nl <= 0 || nh <= 0) return 0;
Multiplicative_dirichlet_convolution_acc<ll> M(nl);
vl al(nl + 1), Ah(nh + 1);
ll x = 1;
repi(i, 1, nl) {
if (i == x * x) {
al[i] = 1;
x++;
}
}
while (x * x >= n / nh) x--;
repir(i, nh, 1) {
while (n / i >= x * x) x++;
Ah[i] = x - 1;
}
//vl a(n + 1);
//rep(i, n) {
// if (i * i > n) break;
// a[i * i] = 1;
//}
//dump(a);
//repi(i, 1, nl) al[i] = a[i];
//repi(i, 2, n) a[i] += a[i - 1];
//dump(a);
//repir(i, nh, 1) Ah[i] = a[n / i];
vl cl(nl + 1, 1), Ch(nh + 1);
cl[0] = 0;
repi(i, 1, nh) Ch[i] = n / i;
//cerr << "- "; repi(i, 1, nl) cerr << i << " \n"[i == nl];
//dump(al); dump(cl);
//cerr << "- "; repi(i, 1, nh) cerr << n / i << " \n"[i == nh];
//dump(Ah); dump(Ch);
M.inv_conv_acc(n, al, Ah, cl, Ch);
return Ch[1];
}
ll solve_N_is_1_TLE(ll L, int n, vl a) {
L /= a[0];
ll nl = (ll)pow(L / 3.31894, 2. / 3) + 1;
chmin(nl, 30043519LL / 2);
ll nh = L / nl + 1;
chmax(nl, nh); chmin(nl, L); chmin(nh, L);
dump(nl); dump(nh);
//cout << "- "; repi(i, 1, nl) cout << i << " \n"[i == nl];
//dump(bl);
//cout << "- "; repi(i, 1, nh) cout << L / i << " \n"[i == nh];
//dump(Bh);
return square_free_sum(L, (int)nl, (int)nh);
}
ll solve_N_is_1(ll n, int hoge, vl a) {
n /= a[0];
vl sq; ll i = 2;
for (; i * i <= n; i++) sq.push_back(i * i);
sq.push_back(i * i);
unordered_map<ll, ll> B;
//
function<ll(ll)> rf = [&](ll n) {
if (B.count(n)) return B[n];
// dump(n);
ll val = n;
repe(i, sq) {
if (i > n) break;
val -= rf(n / i);
}
B[n] = val;
return val;
};
return rf(n);
}
ll solve_TLE(ll n, int hoge, vl a) {
vl sq;
for (ll i = 1; i * i <= n; i++) sq.push_back(i * i);
rep(i, sz(sq)) sq.push_back(2 * sq[i]);
rep(i, sz(sq)) sq.push_back(3 * sq[i]);
sort(all(sq));
sq.erase(sq.begin());
sq.push_back(INFL);
int sqrt_n = (int)sqrt(n) + 1;
vl Bl(sqrt_n + 1), Bh(sqrt_n + 1);
auto B = [&](ll i) { return i < sqrt_n ? Bh[i] : Bl[n / i]; };
//
repi(i, 1, sqrt_n) {
Bl[i] = i;
repe(i2, sq) {
if (i2 > i) break;
Bl[i] -= Bl[i / i2];
}
}
repir(j, sqrt_n, 1) {
ll i = n / j;
Bh[j] = i;
repe(i2, sq) {
if (i2 > i) break;
Bh[j] -= B(i);
}
}
map<ll, int, greater<ll>> tbl;
repe(x, a) tbl[x] = 1;
for (auto it = tbl.begin(); it != tbl.end(); it++) {
auto [i, v] = *it;
tbl[i * 2]++;
tbl[i * 3]++;
tbl[i * 6]++;
}
dump(tbl);
ll res = 0;
for (auto [i, v] : tbl) {
if (i <= n && (v & 1)) res += B(i);
}
return res;
}
#include <boost/unordered_map.hpp>
ll solve(ll n, int hoge, vl a) {
vl sq;
for (ll i = 2; i * i <= n; i++) sq.push_back(i * i);
rep(i, sz(sq)) sq.push_back(2 * sq[i]);
rep(i, sz(sq)) sq.push_back(3 * sq[i]);
sq.push_back(2);
sq.push_back(3);
sq.push_back(6);
sort(all(sq));
sq.push_back(INFL);
constexpr int TH = (int)1e6;
vl B_tbl(TH, -1);
boost::unordered_map<ll, ll> B_map;
auto inB = [&](ll i) { return i < TH ? B_tbl[i] != -1 : B_map.count(i); };
auto getB = [&](ll i) { return i < TH ? B_tbl[i] : B_map[i]; };
auto setB = [&](ll i, ll val) {
if (i < TH) B_tbl[i] = val;
else B_map[i] = val;
};
//
function<ll(ll)> rf = [&](ll n) {
if (inB(n)) return getB(n);
ll val = n;
repe(i, sq) {
if (i > n) break;
val -= rf(n / i);
}
setB(n, val);
return val;
};
map<ll, int, greater<ll>> tbl;
repe(x, a) tbl[x] = 1;
for (auto it = tbl.begin(); it != tbl.end(); it++) {
auto [i, v] = *it;
tbl[i * 2]++;
tbl[i * 3]++;
tbl[i * 6]++;
}
dump(tbl);
ll res = 0;
for (auto [i, v] : tbl) {
if (i <= n && (v & 1)) res += rf(n / i);
}
return res;
}
int main() {
input_from_file("input.txt");
// output_to_file("output.txt");
// zikken();
ll l; int n;
cin >> l >> n;
vl a(n);
cin >> a;
//if (l <= (int)1e5) {
// cout << solve_small_L(l, n, a) << endl;
//}
//else if (n == 1) {
// cout << solve_N_is_1(l, n, a) << endl;
//}
//else {
cout << solve(l, n, a) << endl;
}
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