Theorem: Multiply Perfect Number of Order 6

The number defined as:
:$n = 2^{36} \times 3^8 \times 5^5 \times 7^7 \times 11 \times 13^2 \times 19 \times 31^2$
::$\times \ 43 \times 61 \times 83 \times 223 \times 331 \times 379 \times 601 \times 757 \times 1201$
::$\times \ 7019 \times 112 \, 303 \times 898 \, 423 \times 616 \, 318 \, 177$
is multiply perfect of order $6$.

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} {11}
      | r = 12
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^2 \times 3
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {19}
      | r = 20
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^2 \times 5
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {43}
      | r = 44
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^2 \times 11
      | 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} {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} {223}
      | r = 224
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^5 \times 7
      | 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} {379}
      | r = 380
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^2 \times 5 \times 19
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {601}
      | r = 602
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 7 \times 43
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {757}
      | r = 758
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 379
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {1201}
      | r = 1202
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 601
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {7019}
      | r = 7020
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^2 \times 3^3 \times 5 \times 13
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {112 \, 303}
      | r = 112 \, 304
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^4 \times 7019
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {898 \, 423}
      | r = 898 \, 424
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2^3 \times 112 \, 303
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {616 \, 318 \, 177}
      | r = 616 \, 318 \, 178
      | c = Divisor Sum of Prime Number
}}
{{eqn | r = 2 \times 7^3 \times 898 \, 423
      | 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^{36} }
      | r = 2 \times 2^{36} - 1
      | c = Divisor Sum 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_1} {3^8}
      | r = \dfrac {3^9 - 1} {3 - 1}
      | c = Divisor Sum 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_1} {5^5}
      | r = \dfrac {5^6 - 1} {5 - 1}
      | c = Divisor Sum 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_1} {7^7}
      | r = \dfrac {7^8 - 1} {7 - 1}
      | c = Divisor Sum 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_1} {13^2}
      | r = \dfrac {13^3 - 1} {13 - 1}
      | c = Divisor Sum 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_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}}
Gathering up the prime factors, we have:
:$\map {\sigma_1} n = 2^{37} \times 3^9 \times 5^5 \times 7^7 \times 11 \times 13^2 \times 19 \times 31^2$
::$\times \ 43 \times 61 \times 83 \times 223 \times 331 \times 379 \times 601 \times 757 \times 1201$
::$\times \ 7019 \times 112 \, 303 \times 898 \, 423 \times 616 \, 318 \, 177$
By inspection of the multiplicities of the prime factors of $n$ and $\map {\sigma_1} n$, it can be seen that they match for all except for $2$ and $3$.
It follows that $\map {\sigma_1} n = 2 \times 3 \times n = 6 n$.
Hence the result.
{{qed}}