noip复习——线性筛(欧拉筛)

Posted 2021-11-28 happylittlerabbit

整数的唯一分解定理：

\(\forall A\in \mathbb {N} ,\,A>1\quad \exists \prod _{i=1}^{s}p_{i}^{a_{i}}=A\)，其中\({\displaystyle p_{1}<p_{2}<p_{3}<\cdots <p_{s}}\)而且 \(p_{i}\)是一个质数, \(a_{i}\in \mathbb {Z} ^{+}\)(摘自维基百科)

欧拉筛通过使每个整数只会被它的最小质因子筛到来保证时间复杂度，可以用来筛质数。同时，利用这个性质可以在线性时间内筛出很多积性函数。

---

筛质数

for (int i = 2; i <= n; ++i)
{
    if (!vis[i])
        pri[++cnt] = i;
    for (int j = 1; j <= cnt && pri[j] * i <= n; ++j)
    {
        vis[i * pri[j]] = 1;
        if (i % pri[j] == 0)
            break;
    }
}

---

求欧拉函数\(\varphi\)

for (int i = 2; i <= n; ++i)
{
    if (!vis[i])
        pri[++cnt] = i,
        phi[i] = i - 1;
    for (int j = 1; j <= cnt && pri[j] * i <= n; ++j)
    {
        vis[i * pri[j]] = 1;
        if (i % pri[j])
            phi[i * pri[j]] = phi[i] * phi[pri[j]];
        else
        {
            phi[i * pri[j]] = phi[i] * pri[j];
            break;
        }
    }
}

---

求莫比乌斯函数\(\mu\)

for (int i = 2; i <= n; ++i)
{
    if (!vis[i])
        pri[++cnt] = i,
        mu[i] = -1;
    for (int j = 1; j <= cnt && pri[j] * i <= n; ++j)
    {
        vis[i * pri[j]] = 1;
        if (i % pri[j])
            mu[i * pri[j]] = -mu[i];
        else
            break;
    }
}

---

求约数个数\(\sigma_0(d)\)

for (int i = 2; i <= n; ++i)
{
    if (!vis[i])
        pri[++cnt] = i,
        f[i] = 1,
        d[i] = 2;
    for (int j = 1; j <= cnt && pri[j] * i <= n; ++j)
    {
        vis[i * pri[j]] = 1;
        if (i % pri[j])
            f[i * pri[j]] = 1, d[i * pri[j]] = d[i] * d[pri[j]];
        else
        {
            f[i * pri[j]] = f[i] + 1;
            d[i * pri[j]] = d[i] / (f[i] + 1)* (f[i] + 2);
            break;
        }
    }
}