Divisor Sum of 27,418, 521,963, 671,501, 273,905, 190,135, 082,692, 041,730, 405,303, 870,249, 023,209
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Example of Divisor Sum of Integer
| \(\ds \) | \(\) | \(\ds \map {\sigma_1} {27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 65 \, 400 \, 948 \, 817 \, 364 \, 742 \, 403 \, 487 \, 616 \, 930 \, 512 \, 213 \, 536 \, 407 \, 552 \, 000 \, 000 \, 000 \, 000 \, 000\) |
where $\sigma_1$ denotes the divisor sum.
Proof
We have that:
| \(\ds \) | \(\) | \(\ds 27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 3^9 \times 7^3 \times 11^3 \times 13^3 \times 17^3 \times 41^3 \times 43^3 \times 47^3 \times 443^3 \times 499^3 \times 3583^3\) |
Hence from Divisor Sum of Integer:
| \(\ds \) | \(\) | \(\ds \map {\sigma_1} {27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {3^{10} - 1} {3 - 1} \times \frac {7^4 - 1} {7 - 1} \times \frac {11^4 - 1} {11 - 1} \times \frac {13^4 - 1} {13 - 1} \times \frac {17^4 - 1} {17 - 1} \times \frac {41^4 - 1} {41 - 1} \times \frac {43^4 - 1} {43 - 1} \times \frac {47^4 - 1} {47 - 1} \times \frac {443^4 - 1} {443 - 1} \times \frac {499^4 - 1} {499 - 1} \times \frac {3583^4 - 1} {3583 - 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {59 \, 048} 2 \times \frac {2400} 6 \times \frac {14 \, 640} {10} \times \frac {28 \, 560} {12} \times \frac {83 \, 520} {16} \times \frac {2 \, 825 \, 760} {40} \times \frac {3 \, 418 \, 800} {42}\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds \frac {4 \, 879 \, 680} {46} \times \frac {38 \, 513 \, 670 \, 000} {442} \times \frac {62 \, 001 \, 498 \, 000} {498} \times \frac {164 \, 811 \, 393 \, 976 \, 320} {3582}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 29 \, 524 \times 400 \times 1464 \times 2380 \times 5220 \times 70 \, 644 \times 81 \, 400 \times 106 \, 080 \times 87 \, 135 \, 000 \times 124 \, 501 \, 000 \times 46 \, 010 \, 997 \, 760\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {2^2 \times 11^2 \times 61} \times \paren {2^4 \times 5^2} \times \paren {2^3 \times 3 \times 61} \times \paren {2^2 \times 5 \times 7 \times 17}\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds \paren {2^2 \times 3^2 \times 5 \times 29} \times \paren {2^2 \times 3 \times 7 \times 29^2} \times \paren {2^3 \times 5^2 \times 11 \times 37} \times \paren {2^5 \times 3 \times 5 \times 13 \times 17}\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds \paren {2^3 \times 3 \times 5^4 \times 37 \times 157} \times \paren {2^3 \times 5^3 \times 13 \times 61 \times 157} \times \paren {2^{10} \times 5 \times 7 \times 13 \times 17 \times 37 \times 157}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2^{39} \times 3^6 \times 5^{15} \times 7^3 \times 11^3 \times 13^3 \times 17^3 \times 29^3 \times 37^3 \times 61^3 \times 157^3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 65 \, 400 \, 948 \, 817 \, 364 \, 742 \, 403 \, 487 \, 616 \, 930 \, 512 \, 213 \, 536 \, 407 \, 552 \, 000 \, 000 \, 000 \, 000 \, 000\) |
$\blacksquare$