[sum_{i=1}^{n}sum_{j=1}^{n} ijgcd(i,j)
]
[=sum_{d=1}^{n} d sum_{i=1}^{n}sum_{j=1}^{n} ij[gcd(i,j)==d]
]
[=sum_{d=1}^{n} d^3 sum_{i=1}^{lfloorfrac{n}{d}
floor} sum_{j=1}^{lfloorfrac{n}{d}
floor} ij[gcd(i,j)==1]
]
[=sum_{d=1}^{n} d^3 sum_{i=1}^{lfloorfrac{n}{d}
floor} sum_{j=1}^{lfloorfrac{n}{d}
floor} ij[gcd(i,j)==1]
]
因为 (sum_{t|n} mu(t) = [n == 1])
所以
[=sum_{d=1}^{n} d^3 sum_{i=1}^{lfloorfrac{n}{d}
floor} sum_{j=1}^{lfloorfrac{n}{d}
floor} ij sum_{t|gcd(i,j)} mu(t)
]
[=sum_{d=1}^{n} d^3 sum_{t=1}^{lfloorfrac{n}{d}
floor} mu(t) t^2 sum_{i=1}^{lfloorfrac{n}{td}
floor} sum_{j=1}^{lfloorfrac{n}{td}
floor} ij
]
设 (sum_x = sum_{i=1}^{x})
则原式等同于
[=sum_{d=1}^{n} d^3 sum_{t=1}^{lfloorfrac{n}{d}
floor} mu(t) t^2 (sum_{lfloor frac{n}{td}
floor})^2
]
设 (T=td)
[=sum_{T=1}^{n} T^2 (sum_{lfloor frac{n}{T}
floor})^2 sum_{d|t} mu (frac{T}{d}) d
]
因为
[sum_{d|n} frac{mu(d)}{d} = frac{varphi(n)}{n}
]
根据狄利克雷卷积
[id(x) = x , id * mu = varphi
]
那么这个柿子变成了
[sum_{T=1}^{n}T^2 varphi(T) (sum_{lfloor frac{n}{T}
floor})^2
]
然后我们本身知道 形如 (lfloor frac{n}{T} floor) 之类的玩意可以数论分块
然后 (f_x = x^2 varphi(x)) 的前缀和
设 (s_x) 是 (f_x) 的前缀和，然后考虑 (g) 函数
然后我们知道
[sum_{d|n} varphi(d) = n
]
考虑 (g) 函数为 (g_i = i^2)
[(g*f)(i) = sum_{d|i} d^2 varphi (d) frac{i^2}{d^2} = sum_{d|i}varphi (d) i^2 = i^3
]
[s_n = sum_{i=1}^{n} i^3 - sum_{i=2}^{n} i^2 s_{frac{n}{i}}
]
因为
[sum_{i=1}^{n}i^3 = (sum_{i=1}^{n}i)^2
]
[sum_{i=1}^{n}i^2 = frac{n(n+1)(2n+1)}{6}
]
Q.E.D.
杜教筛一手，这题没了。
// powered by c++11
// by Isaunoya
#include<bits/stdc++.h>
#define rep(i , x , y) for(register int i = (x) ; i <= (y) ; ++ i)
#define Rep(i , x , y) for(register int i = (x) ; i >= (y) ; -- i)
using namespace std ;
using db = double ;
using ll = long long ;
using uint = unsigned int ;
#define int long long
using pii = pair < int , int > ;
#define ve vector
#define Tp template
#define all(v) v.begin() , v.end()
#define sz(v) ((int)v.size())
#define pb emplace_back
#define fir first
#define sec second
// the cmin && cmax
Tp < class T > void cmax(T & x , const T & y) { if(x < y) x = y ; }
Tp < class T > void cmin(T & x , const T & y) { if(x > y) x = y ; }
// sort , unique , reverse
Tp < class T > void sort(ve < T > & v) { sort(all(v)) ; }
Tp < class T > void unique(ve < T > & v) { sort(all(v)) ; v.erase(unique(all(v)) , v.end()) ; }
Tp < class T > void reverse(ve < T > & v) { reverse(all(v)) ; }
const int SZ = 0x191981 ;
struct FILEIN {
~ FILEIN () {} char qwq[SZ] , * S = qwq , * T = qwq , ch ;
char GETC() { return (S == T) && (T = (S = qwq) + fread(qwq , 1 , SZ , stdin) , S == T) ? EOF : * S ++ ; }
FILEIN & operator >> (char & c) { while(isspace(c = GETC())) ; return * this ; }
FILEIN & operator >> (string & s) {
while(isspace(ch = GETC())) ; s = "" + ch ;
while(! isspace(ch = GETC())) s += ch ; return * this ;
}
Tp < class T > void read(T & x) {
bool sign = 1 ; while((ch = GETC()) < 0x30) if(ch == 0x2d) sign = 0 ;
x = (ch ^ 0x30) ; while((ch = GETC()) > 0x2f) x = x * 0xa + (ch ^ 0x30) ;
x = sign ? x : -x ;
}
FILEIN & operator >> (int & x) { return read(x) , * this ; }
FILEIN & operator >> (signed & x) { return read(x) , * this ; }
FILEIN & operator >> (unsigned & x) { return read(x) , * this ; }
} in ;
struct FILEOUT { const static int LIMIT = 0x114514 ;
char quq[SZ] , ST[0x114] ; signed sz , O ;
~ FILEOUT () { sz = O = 0 ; }
void flush() { fwrite(quq , 1 , O , stdout) ; fflush(stdout) ; O = 0 ; }
FILEOUT & operator << (char c) { return quq[O ++] = c , * this ; }
FILEOUT & operator << (string str) {
if(O > LIMIT) flush() ; for(char c : str) quq[O ++] = c ; return * this ;
}
Tp < class T > void write(T x) {
if(O > LIMIT) flush() ; if(x < 0) { quq[O ++] = 0x2d ; x = -x ; }
do { ST[++ sz] = x % 0xa ^ 0x30 ; x /= 0xa ; } while(x) ;
while(sz) quq[O ++] = ST[sz --] ; return ;
}
FILEOUT & operator << (int x) { return write(x) , * this ; }
FILEOUT & operator << (signed x) { return write(x) , * this ; }
FILEOUT & operator << (unsigned x) { return write(x) , * this ; }
} out ;
int n , p ;
const int maxn = 5e6 + 10 ;
int phi[maxn + 5] ;
int iv2 , iv6 ;
int qpow(int x , int y) {
int res = 1 ;
for( ; y ; y >>= 1 , x = x * x % p)
if(y & 1)
res = res * x % p ;
return res ;
}
int sum(int x) {
x = x - (x / p) * p ;
return (x + 1) * x % p * iv2 % p ;
}
int sum2(int x) {
x = x - (x / p) * p ;
return (x + 1) * x % p * (x + x + 1) % p * iv6 % p ;
}
unordered_map < int , int > _phi ;
int s(int x) {
if(x <= maxn) return phi[x] ;
if(_phi[x]) return _phi[x] ;
int ans = sum(x) ;
ans = ans * ans % p ;
for(int l = 2 , r ; l <= x ; l = r + 1) {
r = x / (x / l) ;
ans = (ans - (sum2(r) - sum2(l - 1) + p) % p * s(x / l) % p + p) % p ;
}
return _phi[x] = ans ;
}
signed main() {
#ifdef _WIN64
freopen("testdata.in" , "r" , stdin) ;
#else
ios_base :: sync_with_stdio(false) ;
cin.tie(nullptr) , cout.tie(nullptr) ;
#endif
// code begin.
in >> p >> n ;
iv2 = qpow(2 , p - 2) ;
iv6 = qpow(6 , p - 2) ;
phi[1] = 1 ;
for(int i = 2 ; i <= maxn; i ++) {
if(phi[i])
continue ;
for(int j = i ; j <= maxn ; j += i) {
if(! phi[j])
phi[j] = j ;
phi[j] = phi[j] / i * (i - 1) ;
}
}
for(int i = 1 ; i <= maxn ; i ++)
phi[i] = (phi[i - 1] + 1ll * i * i % p * phi[i] % p) % p ;
int ans = 0 ;
for(int l = 1 , r ; l <= n ; l = r + 1) {
r = n / (n / l) ;
int k = sum(n / l) ;
k = k * k % p ;
ans = (ans + (s(r) - s(l - 1) + p) % p * k % p) % p ;
}
out << ans << '
' ;
return out.flush() , 0 ;
// code end.
}