Divisor Sum of 6232
Jump to navigation
Jump to search
Example of Divisor Sum of Integer
- $\map {\sigma_1} {6232} = 12 \, 600$
where $\sigma_1$ denotes the divisor sum function.
Proof
- $\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
- $6232 = 2^3 \times 19 \times 41$
Hence:
\(\ds \map {\sigma_1} {6232}\) | \(=\) | \(\ds \frac {2^4 - 1} {2 - 1} \times \paren {19 + 1} \times \paren {41 + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {15} 1 \times 20 \times 42\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {3 \times 5} \times \paren {2^2 \times 5} \times \paren {2 \times 3 \times 7}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^3 \times 3^2 \times 5^2 \times 7\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 12 \, 600\) |
$\blacksquare$