Divisor Sum of 510

Example of Divisor Sum of Integer

$\map {\sigma_1} {510} = 1296$

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:

$510 = 2 \times 3 \times 5 \times 17$

Hence:

$\ds \map {\sigma_1} {510}$

$=$

$\ds \paren {2 + 1} \paren {3 + 1} \paren {5 + 1} \paren {17 + 1}$

$\ds $

$=$

$\ds 3 \times 4 \times 6 \times 18$

$\ds $

$=$

$\ds 3 \times 2^2 \times \paren {2 \times 3} \times \paren {2 × 3^2}$

$\ds $

$=$

$\ds 2^4 \times 3^4$

$\ds $

$=$

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

$\ds $

$=$

$\ds 36^2$

$\ds $

$=$

$\ds 1296$

$\blacksquare$