Divisor Sum of 1,705,636

Example of Divisor Sum of Integer

$\map {\sigma_1} {1 \, 705 \, 636} = 2 \, 989 \, 441$

where $\sigma_1$ denotes the divisor sum function.

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

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

We have that:

$1 \, 705 \, 636 = 2^2 \times 653^2$

Hence:

$\ds \map {\sigma_1} {1 \, 705 \, 636}$

$=$

$\ds \frac {2^3 - 1} {2 - 1} \times \frac {653^3 - 1} {653 - 1}$

$\ds $

$=$

$\ds \frac 7 1 \times \frac {278 \, 445 \, 076} {652}$

$\ds $

$=$

$\ds 7 \times 427 \, 063$

$\ds $

$=$

$\ds \paren {7^2 \times 13^2 \times 19^2}^2$

$\ds $

$=$

$\ds 1729^2$

$\ds $

$=$

$\ds 2 \, 989 \, 441$

$\blacksquare$