Divisor Sum of 32,400

Example of Divisor Sum of Integer

$\map {\sigma_1} {32 \, 400} = 116 \, 281$

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:

$32 \, 400 = 2^4 \times 3^4 \times 5^2$

Hence:

$\ds \map {\sigma_1} {32 \, 400}$

$=$

$\ds \frac {2^5 - 1} {2 - 1} \times \frac {3^5 - 1} {3 - 1} \times \frac {5^3 - 1} {5 - 1}$

$\ds $

$=$

$\ds \frac {31} 1 \times \frac {242} 2 \times \frac {124} 4$

$\ds $

$=$

$\ds 31 \times 121 \times 31$

$\ds $

$=$

$\ds \paren {31 \times 11}^2$

$\ds $

$=$

$\ds 341^2$

$\ds $

$=$

$\ds 116 \, 281$

$\blacksquare$