Divisor Sum of 14,182,439,040

Example of Divisor Sum of Integer

$\map {\sigma_1} {14 \, 182 \, 439 \, 040} = 70 \, 912 \, 195 \, 200$

where $\sigma_1$ denotes the divisor sum function.

Proof

We have that:

$14 \, 182 \, 439 \, 040 = 2^7 \times 3^4 \times 5 \times 7 \times 11^2 \times 17 \times 19$

Hence:

\(\ds \map {\sigma_1} {14 \, 182 \, 439 \, 040}\)

\(=\)

\(\ds \frac {2^8 - 1} {2 - 1} \times \frac {3^5 - 1} {3 - 1} \times \paren {5 + 1} \times \paren {7 + 1} \times \frac {11^3 - 1} {11 - 1} \times \paren {17 + 1} \times \paren {19 + 1}\)

Divisor Sum of Integer

\(\ds \)

\(=\)

\(\ds 255 \times \frac {242} 2 \times 6 \times 8 \times \frac {1330} {10} \times 18 \times 20\)

\(\ds \)

\(=\)

\(\ds 255 \times 121 \times 6 \times 8 \times 133 \times 18 \times 20\)

\(\ds \)

\(=\)

\(\ds \paren {3 \times 5 \times 17} \times 11^2 \times \paren {2 \times 3} \times 2^3 \times \paren {7 \times 19} \times \paren {2 \times 3^2} \times \paren {2^2 \times 5}\)

\(\ds \)

\(=\)

\(\ds 2^7 \times 3^4 \times 5^2 \times 7 \times 11^2 \times 17 \times 19\)

\(\ds \)

\(=\)

\(\ds 5 \times \paren {2^7 \times 3^4 \times 5 \times 7 \times 11^2 \times 17 \times 19}\)

\(\ds \)

\(=\)

\(\ds 70 \, 912 \, 195 \, 200\)

$\blacksquare$