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