Divisor Sum of 31,695,652,275

Example of Divisor Sum of Integer

$\map {\sigma_1} {31 \, 695 \, 652 \, 275} = 64 \, 795 \, 852 \, 800$

where $\sigma_1$ denotes the divisor sum function.

Proof

We have that:

$31 \, 695 \, 652 \, 275 = 3 \times 5^2 \times 7 \times 19 \times 53 \times 167 \times 359$

Hence from Divisor Sum of Integer:

\(\ds \map {\sigma_1} {31 \, 695 \, 652 \, 275}\)

\(=\)

\(\ds \paren {3 + 1} \times \frac {5^3 - 1} {5 - 1} \times \paren {7 + 1} \times \paren {19 + 1} \times \paren {53 + 1} \times \paren {167 + 1} \times \paren {359 + 1}\)

\(\ds \)

\(=\)

\(\ds 4 \times \frac {124} 4 \times 8 \times 20 \times 54 \times 168 \times 360\)

\(\ds \)

\(=\)

\(\ds 4 \times 31 \times 8 \times 20 \times 54 \times 168 \times 360\)

\(\ds \)

\(=\)

\(\ds 2^2 \times 31 \times 2^3 \times \paren {2^2 \times 5} \times \paren {2 \times 3^3} \times \paren {2^3 \times 3 \times 7} \times \paren {2^3 \times 3^2 \times 5}\)

\(\ds \)

\(=\)

\(\ds 2^{14} \times 3^6 \times 5^2 \times 7 \times 31\)

\(\ds \)

\(=\)

\(\ds 64 \, 795 \, 852 \, 800\)

$\blacksquare$