cplib-cpp
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Cumulative sum of arithmetic functions （数論的関数の累積和）
(number/arithmetic_cumsum.hpp)
- View this file on GitHub
- Last update: 2022-05-01 16:11:38+09:00
- Include:
#include "number/arithmetic_cumsum.hpp" - Link: 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
$f(s) = \sum_{k=1}^N a_k k^s$ のような形式で表される数論的関数に対して，以下の処理をサポートする．
- 2つの数論的関数の積の計算（$O(N^{2/3} (\log N)^{1/3})$）．
- $\sum_{k=1}^{\lfloor N / i \rfloor} a_k$ の値の取得（$O(1)$）．
なお，一つの関数の保持に $O(\sqrt{N})$ の空間計算量を要する．
使用方法
関数定義
有名な数学関数は専用の関数によって直ちに生成できる．
ゼータ関数
関数 $\sum_{k = 1, \dots} k^x$ （$\mathbf{a} = (1, 1, \dots,)$）は
long long n;
arithmetic_cumsum<mint> z = zeta_cumsum<mint>(n);
によって生成できる．生成に要する計算量は $O(n)$．
ゼータ関数のシフト
$\sum_{i = 1, \dots} i^d i^x$ （$\mathbf{a} = (1^d, 2^d, \dots,)$）は以下で生成できる．与える型は ModInt<> や atcoder::static_modint<> を想定している．
int k;
auto f = zeta_shift_cumsum<mint>(n, k);
生成に要する計算量は $O(nk)$．
約数関数
$\sum_{i = 1, \dots} \sum_{d \mid i} d^k i^x$ は以下で生成できる．
arithmetic_cumsum<mint> d = divisor_func_cumsum<mint>(n, k);
生成に要する計算量は $O(nk)$．
自分で定義した関数
具体的に各項の係数を手で与えるには，以下のようにメンバ変数を設定すればよい．
-
a: 長さ $K$．第 $i$ 要素（$0 \le i < K$）は $a_{i + 1}$ の値に対応する． -
A: 長さ $K$．第 $i$ 要素（$0 \le i < K$）には $\sum_{j = 1}^{i + 1} a_{j}$ の値に対応する（aの累積和を持たせればよい）． -
invA: 長さ $L$．第 $i$ 要素（$0 \le i < L$）には $\sum_{j=1}^{\lfloor N / (i + 1)\rfloor} a_j$ の値を持たせる．
サポートする演算
arithmetic_cumsum<mint> f(N), g(N); // 同一の N をもつもの同士に対してのみ演算が可能．
auto f2 = f + g;
auto f3 = f - g;
auto id = arithmetic_cumsum<mint>::identity(N);
auto h = f * g;
auto i = h / g; // g の a_1 の値が逆元を持てば可能
assert(f == i);
mint t = f.sum(k); // a_1 + ... + a_k を返す． k = floor(N / t) を満たす 1 <= t <= N が存在するような k のみ可能．
問題例
リンク
Depends on
- Lagrange interpolation（多項式の Lagrange 補間）
(formal_power_series/lagrange_interpolation.hpp)
- Sum of exponential times polynomial ($\sum_{i=0}^{N - 1} r^i f(i)$)
(formal_power_series/sum_of_exponential_times_polynomial.hpp)
- Sum of exponential times polynomial limit ($\sum_{i=0}^\infty r^i f(i)$)
(formal_power_series/sum_of_exponential_times_polynomial_limit.hpp)
- Linear sieve （線形篩）
(number/sieve.hpp)
Verified with
Code
#pragma once
#include "../formal_power_series/sum_of_exponential_times_polynomial.hpp"
#include "sieve.hpp"
#include <cassert>
#include <cmath>
#include <vector>
// CUT begin
// Dirichlet product (convolution)
// - a[i] = f(i) (1-origin)
// - A[i - 1] = f(1) + ... + f(i)
// - invA[i - 1] = f(1) + ... + f(N / i)
// Reference:
// 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
template <class T> struct arithmetic_cumsum {
long long N;
int K, L;
bool _verify_shape(const arithmetic_cumsum &x) const {
return N == x.N and K == x.K and L == x.L;
}
std::vector<T> a, A, invA;
arithmetic_cumsum(long long N_) : N(N_) {
K = ceill(std::max(sqrtl(N_), powl(N_ / (N_ > 1 ? log2l(N_) : 1.), 2.0 / 3)));
L = (N_ + K - 1) / K;
a.resize(K), A.resize(K), invA.resize(L);
}
T at(int i) const { // a[i]
assert(i >= 1 and i <= K);
return a[i - 1];
}
T sum(long long i) const { // a[1] + ... + a[i]
assert(i >= 1 and i <= N);
if (i <= K) return A[i - 1];
long long j = N / i;
assert(j >= 1 and j <= L);
return invA[j - 1];
}
arithmetic_cumsum operator+(const arithmetic_cumsum &x) const {
assert(N == x.N and K == x.K and L == x.L);
auto ret = *this;
for (int i = 0; i < ret.K; ++i) {
ret.a[i] += x.a[i];
ret.A[i] += x.A[i];
}
for (int i = 0; i < ret.L; ++i) ret.invA[i] += x.invA[i];
return ret;
}
arithmetic_cumsum operator-(const arithmetic_cumsum &x) const {
assert(N == x.N and K == x.K and L == x.L);
auto ret = *this;
for (int i = 0; i < ret.K; ++i) {
ret.a[i] -= x.a[i];
ret.A[i] -= x.A[i];
}
for (int i = 0; i < ret.L; ++i) ret.invA[i] -= x.invA[i];
return ret;
}
// Dirichlet product (convolution)
arithmetic_cumsum operator*(const arithmetic_cumsum &y) const {
assert(_verify_shape(y));
const arithmetic_cumsum &x = *this;
arithmetic_cumsum ret(x.N);
for (int i = 1; i <= x.K; ++i) {
for (int j = 1; i * j <= x.K; ++j) ret.a[i * j - 1] += x.at(i) * y.at(j);
}
ret.A = ret.a;
for (int i = 1; i < ret.K; i++) ret.A[i] += ret.A[i - 1];
for (int l = 1; l <= x.L; ++l) {
const long long n = x.N / l;
long long i = 1;
for (; i * i <= n; ++i) {
ret.invA[l - 1] += x.at(i) * y.sum(n / i) + x.sum(n / i) * y.at(i);
}
ret.invA[l - 1] -= x.sum(i - 1) * y.sum(i - 1);
}
return ret;
}
bool operator==(const arithmetic_cumsum &y) const {
return _verify_shape(y) and a == y.a and A == y.A and invA == y.invA;
}
arithmetic_cumsum operator/(const arithmetic_cumsum &y) const { // Division (*this) / y
assert(_verify_shape(y));
const arithmetic_cumsum &x = *this;
arithmetic_cumsum ret(x.N);
assert(y.at(1) != 0);
const auto y1inv = 1 / y.at(1);
ret.a = x.a;
for (int i = 1; i <= x.K; ++i) {
ret.a[i - 1] *= y1inv;
for (int j = 2; i * j <= x.K; ++j) ret.a[i * j - 1] -= ret.at(i) * y.at(j);
}
ret.A = ret.a;
for (int i = 2; i <= x.K; ++i) ret.A[i - 1] += ret.A[i - 2];
for (int l = x.L; l; --l) {
const long long n = x.N / l;
if (n <= K) {
ret.invA[l - 1] = ret.sum(n);
} else {
ret.invA[l - 1] = 0;
long long i = 1;
T tmp = 0;
for (; i * i <= n; ++i) {
tmp += y.at(i) * ret.sum(n / i) + y.sum(n / i) * ret.at(i);
}
tmp -= y.sum(i - 1) * ret.sum(i - 1);
ret.invA[l - 1] = (x.sum(n) - tmp) * y1inv;
}
}
return ret;
}
static arithmetic_cumsum identity(long long n) { // [1, 0, ..., 0]
arithmetic_cumsum ret(n);
ret.a[0] = 1;
ret.A.assign(ret.K, 1);
ret.invA.assign(ret.L, 1);
return ret;
}
};
// zeta(s) = [1, 1, 1, ...]
template <class T> arithmetic_cumsum<T> zeta_cumsum(long long n) {
arithmetic_cumsum<T> ret(n);
ret.a.assign(ret.K, 1);
for (int k = 1; k <= ret.K; k++) ret.A[k - 1] = k;
for (int l = 1; l <= ret.L; l++) ret.invA[l - 1] = n / l;
return ret;
}
// zeta(s - 1) = [1, 2, 3, ...]
template <class T> arithmetic_cumsum<T> zeta_shift_1_cumsum(long long n) {
arithmetic_cumsum<T> ret(n);
for (int i = 0; i < ret.K; ++i) {
ret.a[i] = i + 1;
ret.A[i] = T(i + 1) * (i + 2) / 2;
}
for (int l = 1; l <= ret.L; ++l) {
T u = n / l;
ret.invA[l - 1] = u * (u + 1) / 2;
}
return ret;
}
// zeta(s - k) = [1^k, 2^k, 3^k, ...]
template <class T> arithmetic_cumsum<T> zeta_shift_cumsum(long long n, int k) {
if (k == 0) return zeta_cumsum<T>(n);
if (k == 1) return zeta_shift_1_cumsum<T>(n);
arithmetic_cumsum<T> ret(n);
auto init_pows = Sieve(k).enumerate_kth_pows<T>(k, k);
for (int i = 1; i <= ret.K; ++i) {
ret.a[i - 1] = T(i).pow(k);
ret.A[i] = ret.a[i] + (i ? ret.A[i - 1] : 0);
}
for (int l = 0; l < ret.L; ++l) {
ret.invA[l] = sum_of_exponential_times_polynomial<T>(1, init_pows, n / (l + 1) + 1);
}
return ret;
}
// Euler totient phi function phi(s) = zeta(s - 1) / zeta(s)
template <class T> arithmetic_cumsum<T> euler_phi_cumsum(long long n) {
return zeta_shift_1_cumsum<T>(n) / zeta_cumsum<T>(n);
}
// 約数関数の累積和
// sigma_k (n) = \sum_{d \mid n} d^k
template <class T> arithmetic_cumsum<T> divisor_func_cumsum(long long n, int k) {
return zeta_shift_cumsum<T>(n, k) * zeta_cumsum<T>(n);
}
// Cumulative sum of Moebius function (: Mertens function)
template <class T> arithmetic_cumsum<T> moebius_func_cumsum(long long n) {
return arithmetic_cumsum<T>::identity(n) / zeta_cumsum<T>(n);
}#line 2 "formal_power_series/lagrange_interpolation.hpp"
#include <vector>
// CUT begin
// Lagrange interpolation
// Input: [f(0), ..., f(N-1)] (length = N), deg(f) < N
// Output: f(x_eval)
// Complexity: O(N)
// Verified: https://atcoder.jp/contests/arc033/tasks/arc033_4
template <typename MODINT> MODINT interpolate_iota(const std::vector<MODINT> ys, MODINT x_eval) {
const int N = ys.size();
if (x_eval.val() < N) return ys[x_eval.val()];
std::vector<MODINT> facinv(N);
facinv[N - 1] = MODINT(N - 1).fac().inv();
for (int i = N - 1; i > 0; i--) facinv[i - 1] = facinv[i] * i;
std::vector<MODINT> numleft(N);
MODINT numtmp = 1;
for (int i = 0; i < N; i++) {
numleft[i] = numtmp;
numtmp *= x_eval - i;
}
numtmp = 1;
MODINT ret = 0;
for (int i = N - 1; i >= 0; i--) {
MODINT tmp = ys[i] * numleft[i] * numtmp * facinv[i] * facinv[N - 1 - i];
ret += ((N - 1 - i) & 1) ? (-tmp) : tmp;
numtmp *= x_eval - i;
}
return ret;
}
#line 2 "formal_power_series/sum_of_exponential_times_polynomial_limit.hpp"
#include <cassert>
#line 4 "formal_power_series/sum_of_exponential_times_polynomial_limit.hpp"
// CUT begin
// $d$ 次以下の多項式 $f(x)$ と定数 $r$ について，
// $\sum_{i=0}^\infty r^i f(i)$ の値を $[f(0), ..., f(d - 1), f(d)]$ の値から $O(d)$ で計算．
// Requirement: r != 1
// https://judge.yosupo.jp/problem/sum_of_exponential_times_polynomial_limit
// Document: https://hitonanode.github.io/cplib-cpp/formal_power_series/sum_of_exponential_times_polynomial_limit.hpp
template <typename MODINT>
MODINT sum_of_exponential_times_polynomial_limit(MODINT r, std::vector<MODINT> init) {
assert(r != 1);
if (init.empty()) return 0;
if (init.size() == 1) return init[0] / (1 - r);
auto &bs = init;
const int d = int(bs.size()) - 1;
MODINT rp = 1;
for (int i = 1; i <= d; i++) rp *= r, bs[i] = bs[i] * rp + bs[i - 1];
MODINT ret = 0;
rp = 1;
for (int i = 0; i <= d; i++) {
ret += bs[d - i] * MODINT(d + 1).nCr(i) * rp;
rp *= -r;
}
return ret / MODINT(1 - r).pow(d + 1);
};
#line 6 "formal_power_series/sum_of_exponential_times_polynomial.hpp"
// CUT begin
// $d$ 次以下の多項式 $f(x)$ と定数 $r$ について，
// $\sum_{i=0}^{N-1} r^i f(i)$ の値を $[f(0), ..., f(d - 1), f(d)]$ の値から $O(d)$ で計算．
// Reference:
// - https://judge.yosupo.jp/problem/sum_of_exponential_times_polynomial
// - https://min-25.hatenablog.com/entry/2015/04/24/031413
template <typename MODINT>
MODINT sum_of_exponential_times_polynomial(MODINT r, const std::vector<MODINT> &init, long long N) {
if (N <= 0) return 0;
if (init.size() == 1) return init[0] * (r == 1 ? MODINT(N) : (1 - r.pow(N)) / (1 - r));
std::vector<MODINT> S(init.size() + 1);
MODINT rp = 1;
for (int i = 0; i < int(init.size()); i++) {
S[i + 1] = S[i] + init[i] * rp;
rp *= r;
}
if (N < (long long)S.size()) return S[N];
if (r == 1) return interpolate_iota<MODINT>(S, N);
const MODINT Sinf = sum_of_exponential_times_polynomial_limit<MODINT>(r, init);
MODINT rinv = r.inv(), rinvp = 1;
for (int i = 0; i < int(S.size()); i++) {
S[i] = (S[i] - Sinf) * rinvp;
rinvp *= rinv;
}
return interpolate_iota<MODINT>(S, N) * r.pow(N) + Sinf;
};
#line 3 "number/sieve.hpp"
#include <map>
#line 5 "number/sieve.hpp"
// CUT begin
// Linear sieve algorithm for fast prime factorization
// Complexity: O(N) time, O(N) space:
// - MAXN = 10^7: ~44 MB, 80~100 ms (Codeforces / AtCoder GCC, C++17)
// - MAXN = 10^8: ~435 MB, 810~980 ms (Codeforces / AtCoder GCC, C++17)
// Reference:
// [1] D. Gries, J. Misra, "A Linear Sieve Algorithm for Finding Prime Numbers,"
// Communications of the ACM, 21(12), 999-1003, 1978.
// - https://cp-algorithms.com/algebra/prime-sieve-linear.html
// - https://37zigen.com/linear-sieve/
struct Sieve {
std::vector<int> min_factor;
std::vector<int> primes;
Sieve(int MAXN) : min_factor(MAXN + 1) {
for (int d = 2; d <= MAXN; d++) {
if (!min_factor[d]) {
min_factor[d] = d;
primes.emplace_back(d);
}
for (const auto &p : primes) {
if (p > min_factor[d] or d * p > MAXN) break;
min_factor[d * p] = p;
}
}
}
// Prime factorization for 1 <= x <= MAXN^2
// Complexity: O(log x) (x <= MAXN)
// O(MAXN / log MAXN) (MAXN < x <= MAXN^2)
template <class T> std::map<T, int> factorize(T x) const {
std::map<T, int> ret;
assert(x > 0 and
x <= ((long long)min_factor.size() - 1) * ((long long)min_factor.size() - 1));
for (const auto &p : primes) {
if (x < T(min_factor.size())) break;
while (!(x % p)) x /= p, ret[p]++;
}
if (x >= T(min_factor.size())) ret[x]++, x = 1;
while (x > 1) ret[min_factor[x]]++, x /= min_factor[x];
return ret;
}
// Enumerate divisors of 1 <= x <= MAXN^2
// Be careful of highly composite numbers https://oeis.org/A002182/list
// https://gist.github.com/dario2994/fb4713f252ca86c1254d#file-list-txt (n, (# of div. of n)):
// 45360->100, 735134400(<1e9)->1344, 963761198400(<1e12)->6720
template <class T> std::vector<T> divisors(T x) const {
std::vector<T> ret{1};
for (const auto p : factorize(x)) {
int n = ret.size();
for (int i = 0; i < n; i++) {
for (T a = 1, d = 1; d <= p.second; d++) {
a *= p.first;
ret.push_back(ret[i] * a);
}
}
}
return ret; // NOT sorted
}
// Euler phi functions of divisors of given x
// Verified: ABC212 G https://atcoder.jp/contests/abc212/tasks/abc212_g
// Complexity: O(sqrt(x) + d(x))
template <class T> std::map<T, T> euler_of_divisors(T x) const {
assert(x >= 1);
std::map<T, T> ret;
ret[1] = 1;
std::vector<T> divs{1};
for (auto p : factorize(x)) {
int n = ret.size();
for (int i = 0; i < n; i++) {
ret[divs[i] * p.first] = ret[divs[i]] * (p.first - 1);
divs.push_back(divs[i] * p.first);
for (T a = divs[i] * p.first, d = 1; d < p.second; a *= p.first, d++) {
ret[a * p.first] = ret[a] * p.first;
divs.push_back(a * p.first);
}
}
}
return ret;
}
// Moebius function Table, (-1)^{# of different prime factors} for square-free x
// return: [0=>0, 1=>1, 2=>-1, 3=>-1, 4=>0, 5=>-1, 6=>1, 7=>-1, 8=>0, ...] https://oeis.org/A008683
std::vector<int> GenerateMoebiusFunctionTable() const {
std::vector<int> ret(min_factor.size());
for (unsigned i = 1; i < min_factor.size(); i++) {
if (i == 1) {
ret[i] = 1;
} else if ((i / min_factor[i]) % min_factor[i] == 0) {
ret[i] = 0;
} else {
ret[i] = -ret[i / min_factor[i]];
}
}
return ret;
}
// Calculate [0^K, 1^K, ..., nmax^K] in O(nmax)
// Note: **0^0 == 1**
template <class MODINT> std::vector<MODINT> enumerate_kth_pows(long long K, int nmax) const {
assert(nmax < int(min_factor.size()));
assert(K >= 0);
if (K == 0) return std::vector<MODINT>(nmax + 1, 1);
std::vector<MODINT> ret(nmax + 1);
ret[0] = 0, ret[1] = 1;
for (int n = 2; n <= nmax; n++) {
if (min_factor[n] == n) {
ret[n] = MODINT(n).pow(K);
} else {
ret[n] = ret[n / min_factor[n]] * ret[min_factor[n]];
}
}
return ret;
}
};
// Sieve sieve((1 << 20));
#line 5 "number/arithmetic_cumsum.hpp"
#include <cmath>
#line 7 "number/arithmetic_cumsum.hpp"
// CUT begin
// Dirichlet product (convolution)
// - a[i] = f(i) (1-origin)
// - A[i - 1] = f(1) + ... + f(i)
// - invA[i - 1] = f(1) + ... + f(N / i)
// Reference:
// 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
template <class T> struct arithmetic_cumsum {
long long N;
int K, L;
bool _verify_shape(const arithmetic_cumsum &x) const {
return N == x.N and K == x.K and L == x.L;
}
std::vector<T> a, A, invA;
arithmetic_cumsum(long long N_) : N(N_) {
K = ceill(std::max(sqrtl(N_), powl(N_ / (N_ > 1 ? log2l(N_) : 1.), 2.0 / 3)));
L = (N_ + K - 1) / K;
a.resize(K), A.resize(K), invA.resize(L);
}
T at(int i) const { // a[i]
assert(i >= 1 and i <= K);
return a[i - 1];
}
T sum(long long i) const { // a[1] + ... + a[i]
assert(i >= 1 and i <= N);
if (i <= K) return A[i - 1];
long long j = N / i;
assert(j >= 1 and j <= L);
return invA[j - 1];
}
arithmetic_cumsum operator+(const arithmetic_cumsum &x) const {
assert(N == x.N and K == x.K and L == x.L);
auto ret = *this;
for (int i = 0; i < ret.K; ++i) {
ret.a[i] += x.a[i];
ret.A[i] += x.A[i];
}
for (int i = 0; i < ret.L; ++i) ret.invA[i] += x.invA[i];
return ret;
}
arithmetic_cumsum operator-(const arithmetic_cumsum &x) const {
assert(N == x.N and K == x.K and L == x.L);
auto ret = *this;
for (int i = 0; i < ret.K; ++i) {
ret.a[i] -= x.a[i];
ret.A[i] -= x.A[i];
}
for (int i = 0; i < ret.L; ++i) ret.invA[i] -= x.invA[i];
return ret;
}
// Dirichlet product (convolution)
arithmetic_cumsum operator*(const arithmetic_cumsum &y) const {
assert(_verify_shape(y));
const arithmetic_cumsum &x = *this;
arithmetic_cumsum ret(x.N);
for (int i = 1; i <= x.K; ++i) {
for (int j = 1; i * j <= x.K; ++j) ret.a[i * j - 1] += x.at(i) * y.at(j);
}
ret.A = ret.a;
for (int i = 1; i < ret.K; i++) ret.A[i] += ret.A[i - 1];
for (int l = 1; l <= x.L; ++l) {
const long long n = x.N / l;
long long i = 1;
for (; i * i <= n; ++i) {
ret.invA[l - 1] += x.at(i) * y.sum(n / i) + x.sum(n / i) * y.at(i);
}
ret.invA[l - 1] -= x.sum(i - 1) * y.sum(i - 1);
}
return ret;
}
bool operator==(const arithmetic_cumsum &y) const {
return _verify_shape(y) and a == y.a and A == y.A and invA == y.invA;
}
arithmetic_cumsum operator/(const arithmetic_cumsum &y) const { // Division (*this) / y
assert(_verify_shape(y));
const arithmetic_cumsum &x = *this;
arithmetic_cumsum ret(x.N);
assert(y.at(1) != 0);
const auto y1inv = 1 / y.at(1);
ret.a = x.a;
for (int i = 1; i <= x.K; ++i) {
ret.a[i - 1] *= y1inv;
for (int j = 2; i * j <= x.K; ++j) ret.a[i * j - 1] -= ret.at(i) * y.at(j);
}
ret.A = ret.a;
for (int i = 2; i <= x.K; ++i) ret.A[i - 1] += ret.A[i - 2];
for (int l = x.L; l; --l) {
const long long n = x.N / l;
if (n <= K) {
ret.invA[l - 1] = ret.sum(n);
} else {
ret.invA[l - 1] = 0;
long long i = 1;
T tmp = 0;
for (; i * i <= n; ++i) {
tmp += y.at(i) * ret.sum(n / i) + y.sum(n / i) * ret.at(i);
}
tmp -= y.sum(i - 1) * ret.sum(i - 1);
ret.invA[l - 1] = (x.sum(n) - tmp) * y1inv;
}
}
return ret;
}
static arithmetic_cumsum identity(long long n) { // [1, 0, ..., 0]
arithmetic_cumsum ret(n);
ret.a[0] = 1;
ret.A.assign(ret.K, 1);
ret.invA.assign(ret.L, 1);
return ret;
}
};
// zeta(s) = [1, 1, 1, ...]
template <class T> arithmetic_cumsum<T> zeta_cumsum(long long n) {
arithmetic_cumsum<T> ret(n);
ret.a.assign(ret.K, 1);
for (int k = 1; k <= ret.K; k++) ret.A[k - 1] = k;
for (int l = 1; l <= ret.L; l++) ret.invA[l - 1] = n / l;
return ret;
}
// zeta(s - 1) = [1, 2, 3, ...]
template <class T> arithmetic_cumsum<T> zeta_shift_1_cumsum(long long n) {
arithmetic_cumsum<T> ret(n);
for (int i = 0; i < ret.K; ++i) {
ret.a[i] = i + 1;
ret.A[i] = T(i + 1) * (i + 2) / 2;
}
for (int l = 1; l <= ret.L; ++l) {
T u = n / l;
ret.invA[l - 1] = u * (u + 1) / 2;
}
return ret;
}
// zeta(s - k) = [1^k, 2^k, 3^k, ...]
template <class T> arithmetic_cumsum<T> zeta_shift_cumsum(long long n, int k) {
if (k == 0) return zeta_cumsum<T>(n);
if (k == 1) return zeta_shift_1_cumsum<T>(n);
arithmetic_cumsum<T> ret(n);
auto init_pows = Sieve(k).enumerate_kth_pows<T>(k, k);
for (int i = 1; i <= ret.K; ++i) {
ret.a[i - 1] = T(i).pow(k);
ret.A[i] = ret.a[i] + (i ? ret.A[i - 1] : 0);
}
for (int l = 0; l < ret.L; ++l) {
ret.invA[l] = sum_of_exponential_times_polynomial<T>(1, init_pows, n / (l + 1) + 1);
}
return ret;
}
// Euler totient phi function phi(s) = zeta(s - 1) / zeta(s)
template <class T> arithmetic_cumsum<T> euler_phi_cumsum(long long n) {
return zeta_shift_1_cumsum<T>(n) / zeta_cumsum<T>(n);
}
// 約数関数の累積和
// sigma_k (n) = \sum_{d \mid n} d^k
template <class T> arithmetic_cumsum<T> divisor_func_cumsum(long long n, int k) {
return zeta_shift_cumsum<T>(n, k) * zeta_cumsum<T>(n);
}
// Cumulative sum of Moebius function (: Mertens function)
template <class T> arithmetic_cumsum<T> moebius_func_cumsum(long long n) {
return arithmetic_cumsum<T>::identity(n) / zeta_cumsum<T>(n);
}