Divisor Sum of 248
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Example of Divisor Sum of Integer
- $\map {\sigma_1} {248} = 480$
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:
- $248 = 2^3 \times 31$
Hence:
\(\ds \map {\sigma_1} {248}\) | \(=\) | \(\ds \frac {2^4 - 1} {2 - 1} \times \frac {31^2 - 1} {31 - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {15} 1 \times \frac {32 \times 30} {30}\) | Difference of Two Squares | |||||||||||
\(\ds \) | \(=\) | \(\ds 15 \times 32\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {3 \times 5} \times 2^5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 480\) |
$\blacksquare$