Table 16 Some convolution formulas

${\sum }_{k=1}^{N-1}\sigma (k){\sigma }_5(N-k)$

$\begin{array}{l}\frac{1}{504}\{20{\sigma }_7(N)+(21\\ -42N){\sigma }_5(N)+\sigma (N)\}\end{array}$

[[3], (3.18)]

${\sum }_{k=1}^{N-1}{\sigma }_3(k){\sigma }_3(N-k)$

$\frac{1}{120}\{{\sigma }_7(N)-{\sigma }_3(N)\}$

[[3], (3.17)]

${\sum }_{k=1}^{N-1}\sigma (k){\sigma }_7(N-k)$

$\begin{array}{l}\frac{1}{480}\{11{\sigma }_9(N)+(20\\ -30N){\sigma }_7(N)-\sigma (N)\}\end{array}$

[[3], (3.28)]

${\sum }_{k=1}^{N-1}{\sigma }_3(k){\sigma }_5(N-k)$

$\begin{array}{l}\frac{1}{5,040}\{11{\sigma }_9(N)-21{\sigma }_5(N)\\ +10{\sigma }_3(N)\}\end{array}$

[[3], (3.27)]

${\sum }_{k=1}^{N-1}{\sigma }_5(k){\sigma }_5(N-k)$

$\begin{array}{l}\frac{3}{691}\{\frac{65}{756}{\sigma }_{11}(N)+\frac{691}{756}{\sigma }_5(N)\\ -\tau (N)\}\end{array}$

[[4], (21)]

${\sum }_{k=1}^{N-1}{\sigma }_1(k){\sigma }_9(N-k)$

$\begin{array}{l}\frac{455}{30,404}{\sigma }_{11}(N)-\frac{N}{20}{\sigma }_9(N)\\ +\frac{1}{24}{\sigma }_9(N)+\frac{1}{264}{\sigma }_1(N)\\ -\frac{36}{3,455}\tau (N)\end{array}$

[[4], (19)]

${\sum }_{k=1}^{N-1}{\sigma }_3(k){\sigma }_7(N-k)$

$\begin{array}{l}\frac{91}{110,560}{\sigma }_{11}(N)-\frac{1}{240}{\sigma }_7(N)\\ -\frac{1}{480}{\sigma }_3(N)+\frac{15}{2,764}\tau (N)\end{array}$

[[4], (20)]

${\sum }_{k<N/2}{\sigma }_5(k){\sigma }_1(N-2k)$

$\begin{array}{l}\frac{1}{2,142}{\sigma }_7(N)+\frac{2}{51}{\sigma }_7(\frac{N}{2})\\ +\frac{1-2N}{24}{\sigma }_5(\frac{N}{2})\\ +\frac{1}{504}{\sigma }_1(N)-\frac{1}{408}b(N)\end{array}$

[[24], Theorem 5.2]

${\sum }_{k<N/2}{\sigma }_1(k){\sigma }_5(N-2k)$

$\begin{array}{l}\frac{1}{102}{\sigma }_7(N)+\frac{32}{1,071}{\sigma }_7(\frac{N}{2})\\ +\frac{1-N}{24}{\sigma }_5(N)\\ +\frac{1}{504}{\sigma }_1(\frac{N}{2})-\frac{1}{102}b(N)\end{array}$

[[24], Theorem 5.2]

${\sum }_{k<N/2}{\sigma }_3(k){\sigma }_3(N-2k)$

$\begin{array}{l}\frac{1}{2,040}{\sigma }_7(N)+\frac{2}{255}{\sigma }_7(\frac{N}{2})\\ -\frac{1}{240}{\sigma }_3(N)\\ -\frac{1}{240}{\sigma }_3(\frac{N}{2})+\frac{1}{272}b(N)\end{array}$

[[24], Theorem 5.2]

${\sum }_{k<N/2}{\sigma }_7(k){\sigma }_1(N-2k)$

$\begin{array}{l}\frac{1}{14,880}{\sigma }_9(N)+\frac{17}{744}{\sigma }_9(\frac{N}{2})\\ +\frac{2-3N}{48}{\sigma }_7(\frac{N}{2})\\ -\frac{1}{480}{\sigma }_1(N)+\frac{1}{496}c(N)\\ +\frac{2}{31}l(N)\end{array}$

[[24], Theorem 6.2]

${\sum }_{k<N/2}{\sigma }_1(k){\sigma }_7(N-2k)$

$\begin{array}{l}\frac{17}{2,976}{\sigma }_9(N)+\frac{8}{465}{\sigma }_9(\frac{N}{2})\\ +\frac{4-3N}{96}{\sigma }_7(N)\\ -\frac{1}{480}{\sigma }_1(\frac{N}{2})-\frac{1}{62}c(N)\\ -\frac{16}{31}l(N)\end{array}$

[[24], Theorem 6.2]

${\sum }_{k<N/2}{\sigma }_5(k){\sigma }_3(N-2k)$

$\begin{array}{l}\frac{1}{31,248}{\sigma }_9(N)+\frac{1}{465}{\sigma }_9(\frac{N}{2})\\ -\frac{1}{240}{\sigma }_5(\frac{N}{2})+\frac{1}{504}{\sigma }_3(N)\\ -\frac{1}{496}c(N)-\frac{2}{31}l(N)\end{array}$

[[24], Theorem 6.2]

${\sum }_{k<N/2}{\sigma }_3(k){\sigma }_5(N-2k)$

$\begin{array}{l}\frac{1}{7,440}{\sigma }_9(N)+\frac{4}{1,953}{\sigma }_9(\frac{N}{2})\\ -\frac{1}{240}{\sigma }_5(N)+\frac{1}{504}{\sigma }_3(\frac{N}{2})\\ +\frac{1}{248}c(N)+\frac{4}{31}l(N)\end{array}$

[[24], Theorem 6.2]

${\sum }_{k<N/2}{\sigma }_5(k){\sigma }_5(N-2k)$

$\begin{array}{l}\frac{1}{174,132}{\sigma }_{11}(N)+\frac{16}{43,533}{\sigma }_{11}(\frac{N}{2})\\ +\frac{1}{504}{\sigma }_5(N)+\frac{1}{504}{\sigma }_5(\frac{N}{2})\\ -\frac{11}{5,528}\tau (N)-\frac{88}{691}\tau (\frac{N}{2})\end{array}$

[[24], Theorem 7.2]

${\sum }_{k<N/2}{\sigma }_9(k){\sigma }_1(N-2k)$

$\begin{array}{l}\frac{1}{91,212}{\sigma }_{11}(N)+\frac{31}{2,073}{\sigma }_{11}(\frac{N}{2})\\ +\frac{5-6N}{120}{\sigma }_9(\frac{N}{2})+\frac{1}{264}{\sigma }_1(N)\\ -\frac{21}{5,528}\tau (N)-\frac{282}{3,455}\tau (\frac{N}{2})\end{array}$

[[24], Theorem 7.2]

${\sum }_{k<N/2}{\sigma }_1(k){\sigma }_9(N-2k)$

$\begin{array}{l}\frac{31}{8,292}{\sigma }_{11}(N)+\frac{256}{22,803}{\sigma }_{11}(\frac{N}{2})\\ +\frac{5-3N}{120}{\sigma }_9(N)+\frac{1}{264}{\sigma }_1(\frac{N}{2})\\ -\frac{141}{6,910}\tau (N)-\frac{2,688}{691}\tau (\frac{N}{2})\end{array}$

[[24], Theorem 7.2]

${\sum }_{k<N/2}{\sigma }_7(k){\sigma }_3(N-2k)$

$\begin{array}{l}\frac{1}{331,680}{\sigma }_{11}(N)+\frac{17}{20,730}{\sigma }_{11}(\frac{N}{2})\\ -\frac{1}{240}{\sigma }_7(\frac{N}{2})-\frac{1}{480}{\sigma }_3(N)\\ +\frac{23}{11,056}\tau (N)+\frac{91}{1,382}\tau (\frac{N}{2})\end{array}$

[[24], Theorem 7.2]

${\sum }_{k<N/2}{\sigma }_3(k){\sigma }_7(N-2k)$

$\begin{array}{l}\frac{17}{331,680}{\sigma }_{11}(N)+\frac{8}{10,365}{\sigma }_{11}(\frac{N}{2})\\ -\frac{1}{240}{\sigma }_7(N)-\frac{1}{480}{\sigma }_3(\frac{N}{2})\\ +\frac{91}{22,112}\tau (N)+\frac{368}{691}\tau (\frac{N}{2})\end{array}$

[[24], Theorem 7.2]