Table 2 Formulas for $\mathit{f}\mathbf{(}\mathit{N}\mathbf{)}$ with odd N

${\sum }_{l+m+n=N}\tilde{\sigma }(l)\tilde{\sigma }(m)\tilde{\sigma }(n)$

$\frac{1}{64}({\sigma }_5(N)+6(1-N){\sigma }_3(N)+(8N^2-12N+3)\sigma (N))$

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

$\frac{1}{12}({\sigma }_3(N)-\sigma (N))$

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

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

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

$\frac{N}{24}({\sigma }_3(N)-\sigma (N))$

${\sum }_{l+m+n=N}\hat{\sigma }(l)\hat{\sigma }(m)\tilde{\sigma }(n)$

$\frac{1}{576}({\sigma }_5(N)+(6N-2){\sigma }_3(N)+(7-12N)\sigma (N))$

${\sum }_{l+m+n=N}\hat{\sigma }(l)\hat{\sigma }(m)\hat{\sigma }(n)$

$\frac{1}{192}({\sigma }_5(N)-2{\sigma }_3(N)+\sigma (N))$