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usaco-guide/content/7_Advanced/Multiplicative.mdx
Benjamin Qi 569b14ff01 freqs
2020-06-26 14:00:32 -04:00

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---
id: multiplicative
title: "Prefix Sums of Multiplicative Functions"
author: Benjamin Qi
description: "?"
frequency: 0
---
https://codeforces.com/blog/entry/54150
## Linear Time Sieve
https://judge.yosupo.jp/problem/enumerate_primes
## Counting Primes
https://judge.yosupo.jp/problem/counting_primes
## Totient Function
https://judge.yosupo.jp/problem/sum_of_totient_function
```cpp
template<int SZ> struct Sieve {
vi pr;
int sp[SZ], phi[SZ]; // smallest prime that divides
Sieve() { // above is faster
memset(sp,0,sizeof sp);
phi[1] = 1;
FOR(i,2,SZ) {
if (sp[i] == 0) {
sp[i] = i, pr.pb(i); phi[i] = i-1;
} trav(p,pr) {
if (p > sp[i] || i*p >= SZ) break;
sp[i*p] = p;
phi[i*p] = (p == sp[i] ? p : p-1)*phi[i];
}
}
}
};
const int HI = 5000000;
Sieve<HI> S;
ll N;
vmi small(HI), big;
int main() {
setIO(); re(N); big.rsz(N/HI+2);
FOR(i,1,HI) small[i] = small[i-1]+S.phi[i];
ROF(i,1,sz(big)) {
ll mx = N/i; big[i] = mi(mx)*(mx+1)/2; // dbg("HUH",i,big[i]);
for (ll fac = 2, nex; fac <= mx; fac = nex) {
ll quo = mx/fac; nex = mx/quo+1;
big[i] -= (nex-fac)*(quo < HI ? small[quo] : big[i*fac]);
}
}
ps(big[1]);
}
```
(project euler)
(topcoder problem)