Divisor Sum of 29,912,035,725

Example of Divisor Sum of Integer

$\map {\sigma_1} {29 \, 912 \, 035 \, 725} = 64 \, 795 \, 852 \, 800$

where $\sigma_1$ denotes the divisor sum function.

Proof

We have that:

$29 \, 912 \, 035 \, 725 = 3^3 \times 5^2 \times 11 \times 13 \times 431 \times 719$

Hence from Divisor Sum of Integer:

\(\ds \map {\sigma_1} {29 \, 912 \, 035 \, 725}\)

\(=\)

\(\ds \frac {3^4 - 1} {3 - 1} \times \frac {5^3 - 1} {5 - 1} \times \paren {11 + 1} \times \paren {13 + 1} \times \paren {431 + 1} \times \paren {719 + 1}\)

\(\ds \)

\(=\)

\(\ds \frac {80} 2 \times \frac {124} 4 \times 12 \times 14 \times 432 \times 720\)

\(\ds \)

\(=\)

\(\ds 40 \times 31 \times 12 \times 14 \times 432 \times 720\)

\(\ds \)

\(=\)

\(\ds \paren {2^3 \times 5} \times 31 \times \paren {2^2 \times 3} \times \paren {2 \times 7} \times \paren {2^4 \times 3^3} \times \paren {2^4 \times 3^2 \times 5}\)

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

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

$\blacksquare$