关于d函数的筛法

线性筛筛\(\sigma\)

线性筛筛\(\sigma_0\)

\(p\)是质数,\(\sigma_0(p)=2\)

对于一个\(i\)，如果\(i\)和\(p\)互质，根据积性函数得\(\sigma_0 (i\ast p)=\sigma_0 (i)\ast \sigma_0 (p)\)

如果\(i\)和\(p\)不互质，那么\(p|i\)

设\(i=\prod_{i=1}^mP_i^{r_i}\)

则\(p\ast i=\prod_{i=2}^mP_i^{r_i}\ast P_1^{r_i+1}\)

\(\frac{i}{p}=\prod_{i=2}^m P_i^{r_i}\ast P_1^{r_i-1}\)

\(\sigma_0(i)=\prod_{i=1}^m(r_i+1)\)

\(\sigma_0(i\ast p)=\prod_{i=2}^m(r_i+1)+(r_1+2)\)

\(\sigma_0(\frac{i}{p}) = \prod_{i=2}^m(r_i+1)+r_1\)

设\(T=\prod_{i=2}^m(r_i+1)\)

\(\sigma_0(i)=T\ast (r_1+1)\)

\(\sigma_0(i\ast p)=T\ast (r_1+2)=\sigma_0(i)+T\)

\(\sigma_0(\frac{i}{p})=T\ast r_1=\sigma_0(i)-T\)

可得\(\sigma_0(i\ast p)=2\ast \sigma_0(i)-\sigma_0(\frac{i}{p})\)

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线性筛筛\(\sigma\)

\(p\)是质数,\(\sigma(p)=p+1\)

对于一个\(i\)，如果\(i\)和\(p\)互质，根据积性函数得\(\sigma(i\ast p)=\sigma(i)\ast \sigma(p)\)

如果\(i\)和\(p\)不互质，那么\(p|i\)

设\(i=\prod_{i=1}^mP_i^{r_i}\)

则\(p_1\ast i=\prod_{i=2}^mP_i^{r_i}\ast P_1^{r_i+1}\)

\(\frac{i}{p}=\prod_{i=2}^m P_i^{r_i}\ast P_1^{r_i-1}\)

\(\sigma_i=\prod_{i=1}^n\frac{p_i^{r_i+1}-1}{p_i-1}\)

\(\sigma_{i\ast p}=\prod_{i=2}^n\frac{p_i^{r_i+1}-1}{p_i-1}\ast \frac{p_i^{r_1+2}-1}{p_1-1}\)

\(\sigma_{\frac{i}{p}}=\prod_{i=2}^n\frac{p_i^{r_i+1}-1}{p_i-1}\ast \frac{p_i^{r_1}-1}{p_1-1}\)

设\(T=\prod_{i=2}^n\frac{p_i^{r_i+1}-1}{p_i-1}\)

\(\sigma_{i}=T\ast \frac{p_i^{r_1+1}-1}{p_1-1}\)

\(\sigma_{i\ast p}=T\ast \frac{p_i^{r_1+2}-1}{p_1-1}=\sigma_i+T\ast p_1^{r_1+1}\)

\(\sigma_{\frac{i}{p}}=T\ast \frac{p_i^{r_1}-1}{p_1-1}=\sigma_i-T\ast p_1^{r_1}\)

两边乘\(p_1\)得到\(\sigma_{\frac{i}{p}}\ast p_1=p_1\ast \sigma_i-T\ast p_1^{r_1+1}\)

后两个式子相加可得\(\sigma_{i\ast p_1}=(p_1+1)\ast \sigma_i-p_1\ast \sigma_{\frac{i}{p}}\)