It is to be demonstrated that these numbers are amicable. From Sigma Function of Integer: :$\displaystyle \map \sigma n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where: :$\displaystyle n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i} =  p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$ is the prime decomposition of $n$. When $k_i = 1$ the individual factor becomes $\dfrac {p_i^2 - 1} {p_i - 1} = \dfrac {\paren {p_i + 1} \paren {p_i - 1} } {p_i - 1} = p_i + 1$. We establish the contributions to the $\sigma$ values of $m_1$ and $m_2$ by taking the prime factors in turn, and extracting the prime factors of each result. First, the elements common to both: {{begin-eqn}} {{eqn | l = \map \sigma 5       | r = 6       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 3       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {7^2}       | r = \dfrac {7^3 - 1} {7 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {343 - 1} 6       | c =  }} {{eqn | r = 57       | c =  }} {{eqn | r = 3 \times 19       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {11^2}       | r = \dfrac {11^3 - 1} {11 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {1331 - 1} {10}       | c =  }} {{eqn | r = 133       | c =  }} {{eqn | r = 7 \times 19       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {13}       | r = 14       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 7       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {17}       | r = 18       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 3^2       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {19^3}       | r = \dfrac {19^4 - 1} {19 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {130 \, 321 - 1} {18}       | c =  }} {{eqn | r = 7240       | c =  }} {{eqn | r = 2^3 \times 5 \times 181       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {23}       | r = 24       | c = Sigma Function of Prime Number }} {{eqn | r = 2^3 \times 3       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {37}       | r = 38       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 19       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {181}       | r = 182       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 7 \times 13       | c =  }} {{end-eqn}} This gives a common factor of both $\sigma$ values of: {{begin-eqn}} {{eqn | r = 6 \times 57 \times 133 \times 14 \times 18 \times 7240 \times 24 \times 38 \times 182       | o =  }} {{eqn | r = 2^{11} \times 3^5 \times 5 \times 7^3 \times 13 \times 19^3 \times 181       | c =  }} {{end-eqn}} The remaining prime factors of $m_1$: {{begin-eqn}} {{eqn | l = \map \sigma {101 \times 8693 \times 19 \, 479 \, 382 \, 229}       | r = \paren {101 + 1} \paren {8693 + 1} \paren {19 \, 479 \, 382 \, 229 + 1}       | c = Sigma Function of Square-Free Integer }} {{eqn | r = \paren {2 \times 3 \times 17} \paren {2 \times 3^3 \times 7 \times 23} \paren {2 \times 3 \times 5 \times 7 \times 11^2 \times 37 \times 20 \, 719}       | c =  }} {{eqn | r = 2^3 \times 3^5 \times 5 \times 7^2 \times 11^2 \times 17 \times 23 \times 37 \times 20 \, 719       | c =  }} {{end-eqn}} The remaining prime factors of $m_2$: {{begin-eqn}} {{eqn | l = \map \sigma {365 \, 147 \times 47 \, 307 \, 071 \, 129}       | r = \paren {365 \, 147 + 1} \paren {47 \, 307 \, 071 \, 129 + 1}       | c = Sigma Function of Square-Free Integer }} {{eqn | r = \paren {2^2 \times 3^4 \times 7^2 \times 23} \paren {2 \times 3 \times 5 \times 11^2 \times 17 \times 37 \times 20 \, 719}       | c =  }} {{eqn | r = 2^3 \times 3^5 \times 5 \times 7^2 \times 11^2 \times 17 \times 23 \times 37 \times 20 \, 719       | c =  }} {{end-eqn}} and so: {{begin-eqn}} {{eqn | l = \map \sigma {m_1} = \map \sigma {m_2}       | r = \paren {2^{11} \times 3^5 \times 5 \times 7^3 \times 13 \times 19^3 \times 181} \paren {2^3 \times 3^5 \times 5 \times 7^2 \times 11^2 \times 17 \times 23 \times 37 \times 20 \, 719}       | c =  }} {{eqn | r = 2^{14} \times 3^{10} \times 5^2 \times 7^5 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 20 \, 719       | c =  }} {{end-eqn}} It remains to be shown that: :$\map \sigma {m_1} = \map \sigma {m_2} = m_1 + m_2$ So: {{begin-eqn}} {{eqn | l = m_1 + m_2       | r = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times \paren {101 \times 8693 \times 19 \, 479 \, 382 \, 229 + 365 \, 147 \times 47 \, 307 \, 071 \, 129}       | c =  }} {{eqn | r = 6921698310331805 \times \paren {17102761241386397 + 17274035101540963}       | c =  }} {{eqn | r = 6921698310331805 \times 34376796342927360       | c =  }} {{eqn | r = 237945813161460881075326612684800       | c =  }} {{eqn | r = 2^{14} \times 3^{10} \times 5^2 \times 7^5 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 20719       | c =  }} {{end-eqn}} {{qed}}