Divisor Sum of 59,554, 936,495, 441,891, 385,123, 332,422, 108,719, 776,971, 992,921, 810,832, 072,976, 105,472
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Example of Divisor Sum of Integer
| \(\ds \) | \(\) | \(\ds \map {\sigma_1} {59 \, 554 \, 936 \, 495 \, 441 \, 891 \, 385 \, 123 \, 332 \, 422 \, 108 \, 719 \, 776 \, 971 \, 992 \, 921 \, 810 \, 832 \, 072 \, 976 \, 105 \, 472}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 119 \, 109 \, 872 \, 990 \, 883 \, 372 \, 429 \, 911 \, 423 \, 693 \, 257 \, 384 \, 721 \, 768 \, 293 \, 781 \, 054 \, 467 \, 384 \, 195 \, 153 \, 920\) |
where $\sigma_1$ denotes the divisor sum function.
Proof
We have that:
| \(\ds \) | \(\) | \(\ds 59 \, 554 \, 936 \, 495 \, 441 \, 891 \, 385 \, 123 \, 332 \, 422 \, 108 \, 719 \, 776 \, 971 \, 992 \, 921 \, 810 \, 832 \, 072 \, 976 \, 105 \, 472\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2^{47} \times 9 \, 288 \, 811 \, 670 \, 405 \, 087 \times 45 \, 556 \, 233 \, 678 \, 753 \, 109 \, 045 \, 286 \, 896 \, 851 \, 222 \, 527\) |
Hence from Divisor Sum of Integer:
| \(\ds \) | \(\) | \(\ds \map {\sigma_1} {59 \, 554 \, 936 \, 495 \, 441 \, 891 \, 385 \, 123 \, 332 \, 422 \, 108 \, 719 \, 776 \, 971 \, 992 \, 921 \, 810 \, 832 \, 072 \, 976 \, 105 \, 472}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2^{48} - 1} {2 - 1} \times \paren {9 \, 288 \, 811 \, 670 \, 405 \, 087 + 1} \times \paren {45 \, 556 \, 233 \, 678 \, 753 \, 109 \, 045 \, 286 \, 896 \, 851 \, 222 \, 527 + 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 281 \, 474 \, 976 \, 710 \, 655 \times 9 \, 288 \, 811 \, 670 \, 405 \, 088 \times 45 \, 556 \, 233 \, 678 \, 753 \, 109 \, 045 \, 286 \, 896 \, 851 \, 222 \, 528\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {3^2 \times 5 \times 7 \times 13 \times 17 \times 97 \times 241 \times 257 \times 673} \times \paren {2^5 \times 3 \times 96 \, 758 \, 454 \, 900 \, 053}\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds \paren {2^{52} \times 3^2 \times 11^2 \times 9 \, 288 \, 811 \, 670 \, 405 \, 087}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2^{57} \times 3^5 \times 5 \times 7 \times 11^2 \times 13 \times 17 \times 97 \times 241 \times 257 \times 673\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds 96 \, 758 \, 454 \, 900 \, 053 \times 9 \, 288 \, 811 \, 670 \, 405 \, 087\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 119 \, 109 \, 872 \, 990 \, 883 \, 372 \, 429 \, 911 \, 423 \, 693 \, 257 \, 384 \, 721 \, 768 \, 293 \, 781 \, 054 \, 467 \, 384 \, 195 \, 153 \, 920\) |
$\blacksquare$