Divisor Sum of 510
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Example of Divisor Sum of Integer
- $\map {\sigma_1} {510} = 1296$
where $\sigma_1$ denotes the divisor sum function.
Proof
From Divisor Sum of Integer: Corollary
- $\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
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:
\(\ds \map {\sigma_1} {510}\) | \(=\) | \(\ds \paren {2 + 1} \paren {3 + 1} \paren {5 + 1} \paren {17 + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 4 \times 6 \times 18\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 2^2 \times \paren {2 \times 3} \times \paren {2 × 3^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^4 \times 3^4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2^2 \times 3^2}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 36^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1296\) |
$\blacksquare$