Divisor Sum of 154,345,556,085,770,649,600

Example of Divisor Sum of Integer

$\map {\sigma_1} {154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600} = 926 \, 073 \, 336 \, 514 \, 623 \, 897 \, 600$

where $\sigma_1$ denotes the divisor sum function.

Proof

We have that:

$154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600 = 2^{15} \times 3^5 \times 5^2 \times 7^2 \times 11 \times 13 \times 17 \times 19 \times 31 \times 43 \times 257$

Hence:

\(\ds \)

\(\)

\(\ds \map {\sigma_1} {154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600}\)

\(\ds \)

\(=\)

\(\ds \frac {2^{16} - 1} {2 - 1} \times \frac {3^6 - 1} {3 - 1} \times \frac {5^3 - 1} {5 - 1} \times \frac {7^3 - 1} {7 - 1} \times \paren {11 + 1} \times \paren {13 + 1}\)

Divisor Sum of Integer

\(\ds \)

\(\)

\(\, \ds \times \, \)

\(\ds \paren {17 + 1} \times \paren {19 + 1} \times \paren {31 + 1} \times \paren {43 + 1} \times \paren {257 + 1}\)

\(\ds \)

\(=\)

\(\ds 65 \, 535 \times \frac {728} 2 \times \frac {124} 4 \times \frac {342} 6 \times 12 \times 14 \times 18 \times 20 \times 32 \times 44 \times 258\)

\(\ds \)

\(=\)

\(\ds 65 \, 535 \times 364 \times 31 \times 57 \times 12 \times 14 \times 18 \times 20 \times 32 \times 44 \times 258\)

\(\ds \)

\(=\)

\(\ds \paren {3 \times 5 \times 17 \times 257} \times \paren {2^2 \times 7 \times 13} \times 31 \times \paren {3 \times 19} \times \paren {2^2 \times 3} \times \paren {2 \times 7}\)

\(\ds \)

\(\)

\(\, \ds \times \, \)

\(\ds \paren {2 \times 3^2} \times \paren {2^2 \times 5} \times 2^5 \times \paren {2^2 \times 11} \times \paren {2 \times 3 \times 43}\)

\(\ds \)

\(=\)

\(\ds 2^{16} \times 3^6 \times 5^2 \times 7^2 \times 11 \times 13 \times 17 \times 19 \times 31 \times 43 \times 257\)

\(\ds \)

\(=\)

\(\ds 6 \times \paren {2^{15} \times 3^5 \times 5^2 \times 7^2 \times 11 \times 13 \times 17 \times 19 \times 31 \times 43 \times 257}\)

\(\ds \)

\(=\)

\(\ds 926 \, 073 \, 336 \, 514 \, 623 \, 897 \, 600\)

$\blacksquare$