Divisor Sum of 103,340,640

Example of Divisor Sum of Integer

$\map {\sigma_1} {103 \, 340 \, 640} = 350 \, 584 \, 416$

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:

$103 \, 340 \, 640 = 2^5 \times 3 \times 5 \times 13 \times 16 \, 561$

Hence:

$\ds \map {\sigma_1} {103 \, 340 \, 640}$

$=$

$\ds \frac {2^6 - 1} {2 - 1} \paren {3 + 1} \paren {5 + 1} \paren {13 + 1} \paren {16 \, 561 + 1}$

$\ds $

$=$

$\ds 63 \times 4 \times 6 \times 14 \times 16 \, 562$

$\ds $

$=$

$\ds \paren {3^2 \times 7} \times 2^2 \times \paren {2 \times 3} \times \paren {2 \times 7} \times \paren {2 \times 7^2 \times 13^2}$

$\ds $

$=$

$\ds 2^5 \times 3^3 \times 7^4 \times 13^2$

$\ds $

$=$

$\ds 350 \, 584 \, 416$

$\blacksquare$