Theorem: Multiply Perfect Number of Order 8

{{tidy|use {{TL|begin-itemize}} or {{TL|begin-eqn}}, whatever ingenuity you like}}
The number defined as:
:$n = 2^{65} \times 3^{23} \times 5^9 \times 7^{12} \times 11^3 \times 13^3 \times 17^2 \times 19^2 \times 23 \times 29^2 \times 31^2$
::${} \times 37 \times 41 \times 53 \times 61 \times 67^2 \times 71^2 \times 73 \times 83 \times 89 \times 103 \times 127 \times 131$
::${} \times 149 \times 211 \times 307 \times 331 \times 463 \times 521 \times 683 \times 709 \times 1279 \times 2141 \times 2557 \times 5113$
::${} \times 6481 \times 10 \, 429 \times 20 \, 857 \times 110 \, 563 \times 599 \, 479 \times 16 \, 148 \, 168 \, 401$
is multiply perfect of order $8$.

Proof:
From Divisor Sum Function is Multiplicative, we may take each prime factor separately and form $\map {\sigma_1} n$ as the product of the divisor sum of each.
Each of the prime factors which occur with multiplicity $1$ will be treated first.
A prime factor $p$ contributes towards the combined $\sigma_1$ a factor $p + 1$.
Hence we have:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {23}
      | r = 24
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^3 \times 3
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {37}
      | r = 38
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 19
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {41}
      | r = 42
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 3 \times 7
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {53}
      | r = 54
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 3^3
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {61}
      | r = 62
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 31
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {73}
      | r = 74
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 37
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {83}
      | r = 84
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^2 \times 3 \times 7
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {89}
      | r = 90
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 3^2 \times 5
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {103}
      | r = 104
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^3 \times 13
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {127}
      | r = 128
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^7
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {131}
      | r = 132
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^2 \times 3 \times 11
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {149}
      | r = 150
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 3 \times 5^2
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {211}
      | r = 212
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^2 \times 53
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {307}
      | r = 308
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^2 \times 7 \times 11
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {331}
      | r = 332
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^2 \times 83
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {463}
      | r = 464
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^4 \times 29
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {521}
      | r = 522
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 3^2 \times 29
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {683}
      | r = 684
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^2 \times 3^2 \times 19
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {709}
      | r = 710
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 5 \times 71
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {1279}
      | r = 1280
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^8 \times 5
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {2141}
      | r = 2142
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 3^2 \times 7 \times 17
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {2557}
      | r = 2558
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 1279
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {5113}
      | r = 5114
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 2557
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {6481}
      | r = 6482
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 7 \times 463
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {10 \, 429}
      | r = 10 \, 430
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 5 \times 7 \times 149
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {20 \, 857}
      | r = 20 \, 858
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 10 \, 429
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {110 \, 563}
      | r = 110 \, 564
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^2 \times 131 \times 211
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {599 \, 479}
      | r = 599 \, 480
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^3 \times 5 \times 7 \times 2141
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {16 \, 148 \, 168 \, 401}
      | r = 16 \, 148 \, 168 \, 402
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 103 \times 709 \times 110 \, 563
      | c = 
}}
{{end-eqn}}
The remaining factors are treated using Divisor Sum of Power of Prime:
:$\map {\sigma_1} {p^k} = \dfrac {p^{k + 1} - 1} {p - 1}$
Thus:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {2^{65} }
      | r = 2 \times 2^{65} - 1
      | c = Divisor Sum 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_1} {3^{23} }
      | r = \dfrac {3^{24} - 1} {3 - 1}
      | c = Divisor Sum 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_1} {5^9}
      | r = \dfrac {5^{10} - 1} {5 - 1}
      | c = Divisor Sum 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_1} {7^{12} }
      | r = \dfrac {7^{13} - 1} {7 - 1}
      | c = Divisor Sum 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_1} {11^3}
      | r = \dfrac {11^4 - 1} {11 - 1}
      | c = Divisor Sum 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_1} {13^3}
      | r = \dfrac {13^4 - 1} {13 - 1}
      | c = Divisor Sum 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_1} {17^2}
      | r = \dfrac {17^3 - 1} {17 - 1}
      | c = Divisor Sum of Power of Prime
}}
{{eqn | r = \dfrac {4913 - 1} {16}
      | c = 
}}
{{eqn | r = 307
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {19^2}
      | r = \dfrac {19^3 - 1} {19 - 1}
      | c = Divisor Sum 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_1} {29^2}
      | r = \dfrac {29^3 - 1} {29 - 1}
      | c = Divisor Sum 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_1} {31^2}
      | r = \dfrac {31^3 - 1} {31 - 1}
      | c = Divisor Sum 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_1} {67^2}
      | r = \dfrac {67^3 - 1} {67 - 1}
      | c = Divisor Sum 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_1} {71^2}
      | r = \dfrac {71^3 - 1} {71 - 1}
      | c = Divisor Sum 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:
:$\map {\sigma_1} n = 2^{68} \times 3^{23} \times 5^9 \times 7^{12} \times 11^3 \times 13^3 \times 17^2 \times 19^2 \times 23 \times 29^2 \times 31^2$
::$\times 37 \times 41 \times 53 \times 61 \times 67^2 \times 71^2 \times 73 \times 83 \times 89 \times 103 \times 127 \times 131$
::$\times 149 \times 211 \times 307 \times 331 \times 463 \times 521 \times 683 \times 709 \times 1279 \times 2141 \times 2557 \times 5113$
::$\times 6481 \times 10 \, 429 \times 20 \, 857 \times 110 \, 563 \times 599 \, 479 \times 16 \, 148 \, 168 \, 401$
By inspection of the multiplicities of the prime factors of $n$ and $\map {\sigma_1} n$, it can be seen that they match for all except for $2$.
It follows that $\map {\sigma_1} n = 2^3 \times n = 8 n$.
Hence the result.
{{qed}}