Table 3 Formulas for ${\mathit{a}}_{\mathbf{1}}\mathbf{\sim }{\mathit{a}}_{\mathbf{12}}$

${\sum }_{m=1}^{x-1}\sigma (m)\sigma (x-m)$

$\frac{1}{12}(5{\sigma }_3(n)+(1-6n)\sigma (n))$

$\frac{1}{12}(p-1)(p+1)(5p-6)$

${\sum }_{m=1}^{x-1}m\sigma (m)\sigma (x-m)$

$\frac{n}{24}(5{\sigma }_3(n)+(1-6n)\sigma (n))$

$\frac{1}{24}p(p-1)(p+1)(5p-6)$

${\sum }_{m=1}^{x-1}\sigma (m){\sigma }_3(n-m)$

$\begin{array}{l}\frac{1}{240}(21{\sigma }_5(n)+(10-30n){\sigma }_3(n)\\ -\sigma (n))\end{array}$

$\begin{array}{l}\frac{1}{240}(p-1)(p+1)\\ \times (21p^3-30p^2+31p-30)\end{array}$

${\sum }_{m<x/2}\sigma (m)\sigma (x-2m)$

$\begin{array}{l}\frac{1}{24}(2{\sigma }_3(n)+(1-3n)\sigma (n)\\ +8{\sigma }_3(n/2)+(1-6n)\sigma (n/2))\end{array}$

$\frac{1}{24}(p-1)(p+1)(2p-3)$

${\sum }_{m<x/2}{\sigma }_3(m)\sigma (x-2m)$

$\begin{array}{l}\frac{1}{240}({\sigma }_5(n)-\sigma (n)+20{\sigma }_5(n/2)\\ +(10-30n){\sigma }_3(n/2))\end{array}$

$\frac{1}{240}(p-1)p(p+1)(p^2+1)$

${\sum }_{m<x/2}\sigma (m){\sigma }_3(x-2m)$

$\begin{array}{l}\frac{1}{240}(5{\sigma }_5(n)+(10-15n){\sigma }_3(n)\\ +16{\sigma }_5(n/2)-\sigma (n/2))\end{array}$

$\frac{1}{48}(p-1)(p+1)(p^3-3p^2+3p-3)$

${\sum }_{l+m+s=x}\tilde{\sigma }(l)\tilde{\sigma }(m)\tilde{\sigma }(s)$

$\begin{array}{l}\frac{1}{64}\{{\tilde{\sigma }}_5(n)+6(1-n){\tilde{\sigma }}_3(n)\\ +(8n^2-12n+3)\tilde{\sigma }(n)\}\end{array}$

$\frac{1}{64}(p+1){(p-1)}^2(p^2-5p+10)$

${\sum }_{m=1}^{x-1}\hat{\sigma }(m)\hat{\sigma }(x-m)$

$\frac{1}{12}({\sigma }_3(n)+4{\sigma }_3(\frac{n}{2})-\hat{\sigma }(n))$

$\frac{1}{12}(p-1)p(p+1)$

${\sum }_{m=1}^{x-1}\hat{\sigma }(m){\tilde{\sigma }}_3(x-m)$

$-\frac{1}{48}({\tilde{\sigma }}_5(n)+2{\tilde{\sigma }}_3(n)-3\hat{\sigma }(n))$

$-\frac{1}{48}(p-1)p(p+1)(p^2+3)$

${\sum }_{m=1}^{x-1}m\hat{\sigma }(m)\hat{\sigma }(x-m)$

$\frac{n}{24}({\sigma }_3(n)+4{\sigma }_3(\frac{n}{2})-\hat{\sigma }(n))$

$\frac{1}{24}(p-1)p^2(p+1)$

${\sum }_{l+m+s=x}\hat{\sigma }(l)\hat{\sigma }(m)\tilde{\sigma }(s)$

$\begin{array}{l}\frac{1}{576}\{{\tilde{\sigma }}_5(n)+4{\tilde{\sigma }}_3(n)+\tilde{\sigma }(n)\\ -6(2n-1)\hat{\sigma }(n)\\ +6(n-1)({\sigma }_3(n)+4{\sigma }_3(\frac{n}{2}))\}\end{array}$

$\frac{1}{576}{(p-1)}^2{(p+1)}^2(p+6)$

${\sum }_{l+m+s=x}\hat{\sigma }(l)\hat{\sigma }(m)\hat{\sigma }(s)$

$\begin{array}{l}\frac{1}{192}({\sigma }_5(n)-8{\sigma }_5(\frac{n}{2})-2{\sigma }_3(n)\\ -8{\sigma }_3(\frac{n}{2})+\hat{\sigma }(n))\end{array}$

$\frac{1}{192}{(p-1)}^{2}p{(p+1)}^2$