Divisor Sum of 1680
Jump to navigation
Jump to search
Example of Divisor Sum of Integer
- $\map {\sigma_1} {1680} = 5952$
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:
- $1680 = 2^4 \times 3 \times 5 \times 7$
Hence:
\(\ds \map {\sigma_1} {1680}\) | \(=\) | \(\ds \frac {2^5 - 1} {2 - 1} \times \paren {3 + 1} \times \paren {5 + 1} \times \paren {7 + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {31} 1 \times 4 \times 6 \times 8\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 31 \times 2^2 \times \times \paren {2 \times 3} \times 2^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^6 \times 3 \times 31\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 5952\) |
$\blacksquare$