Divisor Sum of 630
Jump to navigation
Jump to search
Example of Divisor Sum of Integer
- $\map {\sigma_1} {630} = 1872$
where $\sigma_1$ denotes the divisor sum function.
Proof
From Divisor Sum of Integer:
- $\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
- $630 = 2 \times 3^2 \times 5 \times 7$
Hence:
\(\ds \map {\sigma_1} {630}\) | \(=\) | \(\ds \paren {2 + 1} \times \frac {3^3 - 1} {3 - 1} \times \paren {5 + 1} \times \paren {7 + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 13 \times 6 \times 8\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 13 \times \paren {2 \times 3} \times 2^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^4 \times 3^2 \times 13\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1872\) |
$\blacksquare$