From Sigma Function is Multiplicative, we may take each prime factor separately and form $\sigma \left({n}\right)$ as the product of the $\sigma$ function of each. Each of the prime factors which occur with multiplicity $1$ will be treated first. A prime factor $p$ contributes towards the combined $\sigma$ a factor $p + 1$. Hence we have: {{begin-eqn}} {{eqn | l = \map \sigma {11}       | r = 12       | c = Sigma Function of Prime Number }} {{eqn | r = 2^2 \times 3       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {19}       | r = 20       | c = Sigma Function of Prime Number }} {{eqn | r = 2^2 \times 5       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {43}       | r = 44       | c = Sigma Function of Prime Number }} {{eqn | r = 2^2 \times 11       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {61}       | r = 62       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 31       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {83}       | r = 84       | c = Sigma Function of Prime Number }} {{eqn | r = 2^2 \times 3 \times 7       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {223}       | r = 224       | c = Sigma Function of Prime Number }} {{eqn | r = 2^5 \times 7       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {331}       | r = 332       | c = Sigma Function of Prime Number }} {{eqn | r = 2^2 \times 83       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {379}       | r = 380       | c = Sigma Function of Prime Number }} {{eqn | r = 2^2 \times 5 \times 19       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {601}       | r = 602       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 7 \times 43       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {757}       | r = 758       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 379       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {1201}       | r = 1202       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 601       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {7019}       | r = 7020       | c = Sigma Function of Prime Number }} {{eqn | r = 2^2 \times 3^3 \times 5 \times 13       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {112 \, 303}       | r = 112 \, 304       | c = Sigma Function of Prime Number }} {{eqn | r = 2^4 \times 7019       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {898 \, 423}       | r = 898 \, 424       | c = Sigma Function of Prime Number }} {{eqn | r = 2^3 \times 112 \, 303       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {616 \, 318 \, 177}       | r = 616 \, 318 \, 178       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 7^3 \times 898 \, 423       | c =  }} {{end-eqn}} The remaining factors are treated using Sigma Function of Power of Prime: : $\map \sigma {p^k} = \dfrac {p^{k + 1} - 1} {p - 1}$ Thus: {{begin-eqn}} {{eqn | l = \map \sigma {2^{36} }       | r = 2 \times 2^{36} - 1       | c = Sigma Function of Power of 2 }} {{eqn | r = 137 \, 438 \, 953 \, 471       | c =  }} {{eqn | r = 223 \times 616 \, 318 \, 177       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {3^8}       | r = \dfrac {3^9 - 1} {3 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {19 \, 683 - 1} 2       | c =  }} {{eqn | r = 9841       | c =  }} {{eqn | r = 13 \times 757       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {5^5}       | r = \dfrac {5^6 - 1} {5 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {15 \, 625 - 1} 4       | c =  }} {{eqn | r = 3906       | c =  }} {{eqn | r = 2 \times 3^2 \times 7 \times 31       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {7^7}       | r = \dfrac {7^8 - 1} {7 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {5 \, 764 \, 801 - 1} 6       | c =  }} {{eqn | r = 960 \, 800       | c =  }} {{eqn | r = 2^5 \times 5^2 \times 1201       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {13^2}       | r = \dfrac {13^3 - 1} {13 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {2197 - 1} {12}       | c =  }} {{eqn | r = 183       | c =  }} {{eqn | r = 3 \times 61       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {31^2}       | r = \dfrac {31^3 - 1} {31 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {29 \, 791 - 1} {30}       | c =  }} {{eqn | r = 993       | c =  }} {{eqn | r = 3 \times 331       | c =  }} {{end-eqn}} Gathering up the prime factors, we have: :$\map \sigma n = 2^{37} \times 3^9 \times 5^5 \times 7^7 \times 11 \times 13^2 \times 19 \times 31^2$ ::$\times \ 43 \times 61 \times 83 \times 223 \times 331 \times 379 \times 601 \times 757 \times 1201$ ::$\times \ 7019 \times 112 \, 303 \times 898 \, 423 \times 616 \, 318 \, 177$ By inspection of the multiplicities of the prime factors of $n$ and $\map \sigma n$, it can be seen that they match for all except for $2$ and $3$. It follows that $\map \sigma n = 2 \times 3 \times n = 6 n$. Hence the result. {{qed}}