Multiply Perfect Number of Order 8

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Theorem

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:

\(\ds \map {\sigma_1} {23}\) \(=\) \(\ds 24\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2^3 \times 3\)


\(\ds \map {\sigma_1} {37}\) \(=\) \(\ds 38\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2 \times 19\)


\(\ds \map {\sigma_1} {41}\) \(=\) \(\ds 42\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2 \times 3 \times 7\)


\(\ds \map {\sigma_1} {53}\) \(=\) \(\ds 54\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2 \times 3^3\)


\(\ds \map {\sigma_1} {61}\) \(=\) \(\ds 62\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2 \times 31\)


\(\ds \map {\sigma_1} {73}\) \(=\) \(\ds 74\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2 \times 37\)


\(\ds \map {\sigma_1} {83}\) \(=\) \(\ds 84\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2^2 \times 3 \times 7\)


\(\ds \map {\sigma_1} {89}\) \(=\) \(\ds 90\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2 \times 3^2 \times 5\)


\(\ds \map {\sigma_1} {103}\) \(=\) \(\ds 104\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2^3 \times 13\)


\(\ds \map {\sigma_1} {127}\) \(=\) \(\ds 128\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2^7\)


\(\ds \map {\sigma_1} {131}\) \(=\) \(\ds 132\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2^2 \times 3 \times 11\)


\(\ds \map {\sigma_1} {149}\) \(=\) \(\ds 150\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2 \times 3 \times 5^2\)


\(\ds \map {\sigma_1} {211}\) \(=\) \(\ds 212\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2^2 \times 53\)


\(\ds \map {\sigma_1} {307}\) \(=\) \(\ds 308\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2^2 \times 7 \times 11\)


\(\ds \map {\sigma_1} {331}\) \(=\) \(\ds 332\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2^2 \times 83\)


\(\ds \map {\sigma_1} {463}\) \(=\) \(\ds 464\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2^4 \times 29\)


\(\ds \map {\sigma_1} {521}\) \(=\) \(\ds 522\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2 \times 3^2 \times 29\)


\(\ds \map {\sigma_1} {683}\) \(=\) \(\ds 684\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2^2 \times 3^2 \times 19\)


\(\ds \map {\sigma_1} {709}\) \(=\) \(\ds 710\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2 \times 5 \times 71\)


\(\ds \map {\sigma_1} {1279}\) \(=\) \(\ds 1280\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2^8 \times 5\)


\(\ds \map {\sigma_1} {2141}\) \(=\) \(\ds 2142\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2 \times 3^2 \times 7 \times 17\)


\(\ds \map {\sigma_1} {2557}\) \(=\) \(\ds 2558\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2 \times 1279\)


\(\ds \map {\sigma_1} {5113}\) \(=\) \(\ds 5114\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2 \times 2557\)


\(\ds \map {\sigma_1} {6481}\) \(=\) \(\ds 6482\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2 \times 7 \times 463\)


\(\ds \map {\sigma_1} {10 \, 429}\) \(=\) \(\ds 10 \, 430\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2 \times 5 \times 7 \times 149\)


\(\ds \map {\sigma_1} {20 \, 857}\) \(=\) \(\ds 20 \, 858\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2 \times 10 \, 429\)


\(\ds \map {\sigma_1} {110 \, 563}\) \(=\) \(\ds 110 \, 564\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2^2 \times 131 \times 211\)


\(\ds \map {\sigma_1} {599 \, 479}\) \(=\) \(\ds 599 \, 480\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2^3 \times 5 \times 7 \times 2141\)


\(\ds \map {\sigma_1} {16 \, 148 \, 168 \, 401}\) \(=\) \(\ds 16 \, 148 \, 168 \, 402\) Divisor Sum of Prime Number
\(\ds \) \(=\) \(\ds 2 \times 103 \times 709 \times 110 \, 563\)


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:

\(\ds \map {\sigma_1} {2^{65} }\) \(=\) \(\ds 2 \times 2^{65} - 1\) Divisor Sum of Power of 2
\(\ds \) \(=\) \(\ds 73 \, 786 \, 976 \, 294 \, 838 \, 206 \, 463\)
\(\ds \) \(=\) \(\ds 3^2 \times 7 \times 23 \times 67 \times 89 \times 683 \times 20 \, 857 \times 599 \, 479\)


\(\ds \map {\sigma_1} {3^{23} }\) \(=\) \(\ds \dfrac {3^{24} - 1} {3 - 1}\) Divisor Sum of Power of Prime
\(\ds \) \(=\) \(\ds \dfrac {282 \, 429 \, 536 \, 481 - 1} 2\)
\(\ds \) \(=\) \(\ds 141 \, 214 \, 768 \, 240\)
\(\ds \) \(=\) \(\ds 2^4 \times 5 \times 7 \times 13 \times 41 \times 73 \times 6481\)


\(\ds \map {\sigma_1} {5^9}\) \(=\) \(\ds \dfrac {5^{10} - 1} {5 - 1}\) Divisor Sum of Power of Prime
\(\ds \) \(=\) \(\ds \dfrac {9 \, 765 \, 625 - 1} 4\)
\(\ds \) \(=\) \(\ds 2 \, 441 \, 406\)
\(\ds \) \(=\) \(\ds 2 \times 3 \times 11 \times 71 \times 521\)


\(\ds \map {\sigma_1} {7^{12} }\) \(=\) \(\ds \dfrac {7^{13} - 1} {7 - 1}\) Divisor Sum of Power of Prime
\(\ds \) \(=\) \(\ds \dfrac {96 \, 889 \, 010 \, 407 - 1} 6\)
\(\ds \) \(=\) \(\ds 16 \, 148 \, 168 \, 401\)


\(\ds \map {\sigma_1} {11^3}\) \(=\) \(\ds \dfrac {11^4 - 1} {11 - 1}\) Divisor Sum of Power of Prime
\(\ds \) \(=\) \(\ds \dfrac {14 \, 641 - 1} {10}\)
\(\ds \) \(=\) \(\ds 1464\)
\(\ds \) \(=\) \(\ds 2^3 \times 3 \times 61\)


\(\ds \map {\sigma_1} {13^3}\) \(=\) \(\ds \dfrac {13^4 - 1} {13 - 1}\) Divisor Sum of Power of Prime
\(\ds \) \(=\) \(\ds \dfrac {28 \, 561 - 1} {12}\)
\(\ds \) \(=\) \(\ds 2380\)
\(\ds \) \(=\) \(\ds 2^2 \times 5 \times 7 \times 17\)


\(\ds \map {\sigma_1} {17^2}\) \(=\) \(\ds \dfrac {17^3 - 1} {17 - 1}\) Divisor Sum of Power of Prime
\(\ds \) \(=\) \(\ds \dfrac {4913 - 1} {16}\)
\(\ds \) \(=\) \(\ds 307\)


\(\ds \map {\sigma_1} {19^2}\) \(=\) \(\ds \dfrac {19^3 - 1} {19 - 1}\) Divisor Sum of Power of Prime
\(\ds \) \(=\) \(\ds \dfrac {6859 - 1} {18}\)
\(\ds \) \(=\) \(\ds 381\)
\(\ds \) \(=\) \(\ds 3 \times 127\)


\(\ds \map {\sigma_1} {29^2}\) \(=\) \(\ds \dfrac {29^3 - 1} {29 - 1}\) Divisor Sum of Power of Prime
\(\ds \) \(=\) \(\ds \dfrac {24 \, 389 - 1} {28}\)
\(\ds \) \(=\) \(\ds 871\)
\(\ds \) \(=\) \(\ds 13 \times 67\)


\(\ds \map {\sigma_1} {31^2}\) \(=\) \(\ds \dfrac {31^3 - 1} {31 - 1}\) Divisor Sum of Power of Prime
\(\ds \) \(=\) \(\ds \dfrac {29 \, 791 - 1} {30}\)
\(\ds \) \(=\) \(\ds 993\)
\(\ds \) \(=\) \(\ds 3 \times 331\)


\(\ds \map {\sigma_1} {67^2}\) \(=\) \(\ds \dfrac {67^3 - 1} {67 - 1}\) Divisor Sum of Power of Prime
\(\ds \) \(=\) \(\ds \dfrac {300 \, 763 - 1} {66}\)
\(\ds \) \(=\) \(\ds 4557\)
\(\ds \) \(=\) \(\ds 3 \times 7^2 \times 31\)


\(\ds \map {\sigma_1} {71^2}\) \(=\) \(\ds \dfrac {71^3 - 1} {71 - 1}\) Divisor Sum of Power of Prime
\(\ds \) \(=\) \(\ds \dfrac {357 \, 911 - 1} {66}\)
\(\ds \) \(=\) \(\ds 5113\)


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.

$\blacksquare$


Sources