Divisor Sum of 8,212,890,625

Example of Divisor Sum of Integer

$\map {\sigma_1} {8 \, 212 \, 890 \, 625} = 10 \, 632 \, 324 \, 001$

where $\sigma_1$ denotes the divisor sum.

$\ds \map {\sigma_1} n = \prod_{i \mathop = 1}^r \frac {p_i^{k_i + 1} - 1} {p_i - 1}$

where $n = \ds \prod_{1 \mathop = 1}^r p_i^{k_i}$ denotes the prime decomposition of $n$.

We have that:

$8 \, 212 \, 890 \, 625 = 5^{10} \times 29^2$

Hence:

$\ds \map {\sigma_1} {8 \, 212 \, 890 \, 625}$

$=$

$\ds \frac {5^11 - 1} {5 - 1} \times \frac {29^3 - 1} {29 - 1}$

$\ds $

$=$

$\ds \frac {48 \, 828 \, 124} 4 \times \frac {24 \, 388} {28}$

$\ds $

$=$

$\ds 12 \, 207 \, 031 \times 871$

$\ds $

$=$

$\ds 12 \, 207 \, 031 \times \paren {13 \times 67}$

$\ds $

$=$

$\ds 10 \, 632 \, 324 \, 001$

$\blacksquare$