From Sigma Function is Multiplicative, we may take each prime factor separately and form $\map \sigma n$ 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 {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 {41}       | r = 42       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 3 \times 7       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {53}       | r = 54       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 3^3       | 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 {73}       | r = 74       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 37       | 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 {89}       | r = 90       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 3^2 \times 5       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {103}       | r = 104       | c = Sigma Function of Prime Number }} {{eqn | r = 2^3 \times 13       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {127}       | r = 128       | c = Sigma Function of Prime Number }} {{eqn | r = 2^7       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {131}       | r = 132       | c = Sigma Function of Prime Number }} {{eqn | r = 2^2 \times 3 \times 11       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {149}       | r = 150       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 3 \times 5^2       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {211}       | r = 212       | c = Sigma Function of Prime Number }} {{eqn | r = 2^2 \times 53       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {307}       | r = 308       | c = Sigma Function of Prime Number }} {{eqn | r = 2^2 \times 7 \times 11       | 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 {463}       | r = 464       | c = Sigma Function of Prime Number }} {{eqn | r = 2^4 \times 29       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {521}       | r = 522       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 3^2 \times 29       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {683}       | r = 684       | c = Sigma Function of Prime Number }} {{eqn | r = 2^2 \times 3^2 \times 19       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {709}       | r = 710       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 5 \times 71       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {1279}       | r = 1280       | c = Sigma Function of Prime Number }} {{eqn | r = 2^8 \times 5       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {2141}       | r = 2142       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 3^2 \times 7 \times 17       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {2557}       | r = 2558       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 1279       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {5113}       | r = 5114       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 2557       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {6481}       | r = 6482       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 7 \times 463       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {10 \, 429}       | r = 10 \, 430       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 5 \times 7 \times 149       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {20 \, 857}       | r = 20 \, 858       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 10 \, 429       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {110 \, 563}       | r = 110 \, 564       | c = Sigma Function of Prime Number }} {{eqn | r = 2^2 \times 131 \times 211       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {599 \, 479}       | r = 599 \, 480       | c = Sigma Function of Prime Number }} {{eqn | r = 2^3 \times 5 \times 7 \times 2141       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {16 \, 148 \, 168 \, 401}       | r = 16 \, 148 \, 168 \, 402       | c = Sigma Function of Prime Number }} {{eqn | r = 2 \times 103 \times 709 \times 110 \, 563       | 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^{65} }       | r = 2 \times 2^{65} - 1       | c = Sigma Function of Power of 2 }} {{eqn | r = 73 \, 786 \, 976 \, 294 \, 838 \, 206 \, 463       | c =  }} {{eqn | r = 3^2 \times 7 \times 23 \times 67 \times 89 \times 683 \times 20 \, 857 \times 599 \, 479       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {3^{23} }       | r = \dfrac {3^{24} - 1} {3 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {282 \, 429 \, 536 \, 481 - 1} 2       | c =  }} {{eqn | r = 141 \, 214 \, 768 \, 240       | c =  }} {{eqn | r = 2^4 \times 5 \times 7 \times 13 \times 41 \times 73 \times 6481       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {5^9}       | r = \dfrac {5^{10} - 1} {5 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {9 \, 765 \, 625 - 1} 4       | c =  }} {{eqn | r = 2 \, 441 \, 406       | c =  }} {{eqn | r = 2 \times 3 \times 11 \times 71 \times 521       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {7^{12} }       | r = \dfrac {7^{13} - 1} {7 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {96 \, 889 \, 010 \, 407 - 1} 6       | c =  }} {{eqn | r = 16 \, 148 \, 168 \, 401       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {11^3}       | r = \dfrac {11^4 - 1} {11 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {14 \, 641 - 1} {10}       | c =  }} {{eqn | r = 1464       | c =  }} {{eqn | r = 2^3 \times 3 \times 61       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {13^3}       | r = \dfrac {13^4 - 1} {13 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {28 \, 561 - 1} {12}       | c =  }} {{eqn | r = 2380       | c =  }} {{eqn | r = 2^2 \times 5 \times 7 \times 17       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {17^2}       | r = \dfrac {17^3 - 1} {17 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {4913 - 1} {16}       | c =  }} {{eqn | r = 307       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {19^2}       | r = \dfrac {19^3 - 1} {19 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {6859 - 1} {18}       | c =  }} {{eqn | r = 381       | c =  }} {{eqn | r = 3 \times 127       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {29^2}       | r = \dfrac {29^3 - 1} {29 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {24 \, 389 - 1} {28}       | c =  }} {{eqn | r = 871       | c =  }} {{eqn | r = 13 \times 67       | 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}} {{begin-eqn}} {{eqn | l = \map \sigma {67^2}       | r = \dfrac {67^3 - 1} {67 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {300 \, 763 - 1} {66}       | c =  }} {{eqn | r = 4557       | c =  }} {{eqn | r = 3 \times 7^2 \times 31       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \sigma {71^2}       | r = \dfrac {71^3 - 1} {71 - 1}       | c = Sigma Function of Power of Prime }} {{eqn | r = \dfrac {357 \, 911 - 1} {66}       | c =  }} {{eqn | r = 5113       | c =  }} {{end-eqn}} Gathering up the prime factors, we have: :$\sigma \left({n}\right) = 2^{68} \times 3^{23} \times 5^9 \times 7^{12} \times 11^3 \times 13^3 \times 17^2 \times 19^2 \times 23 \times 29^2 \times 31^2$ ::$\times 37 \times 41 \times 53 \times 61 \times 67^2 \times 71^2 \times 73 \times 83 \times 89 \times 103 \times 127 \times 131$ ::$\times 149 \times 211 \times 307 \times 331 \times 463 \times 521 \times 683 \times 709 \times 1279 \times 2141 \times 2557 \times 5113$ ::$\times 6481 \times 10 \, 429 \times 20 \, 857 \times 110 \, 563 \times 599 \, 479 \times 16 \, 148 \, 168 \, 401$ 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 $2$. It follows that $\map \sigma n = 2^3 \times n = 8 n$. Hence the result. {{qed}}