Divisor Sum of 18,416

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Example of Divisor Sum of Integer

$\map {\sigma_1} {18 \, 416} = 35 \, 712$

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:

$18 \, 416 = 2^4 \times 1151$


Hence:

\(\ds \map {\sigma_1} {18 \, 416}\) \(=\) \(\ds \frac {2^5 - 1} {2 - 1} \times \paren {1151 + 1}\)
\(\ds \) \(=\) \(\ds \frac {31} 1 \times 1152\)
\(\ds \) \(=\) \(\ds 31 \times \paren {2^7 \times 3^2}\)
\(\ds \) \(=\) \(\ds 2^7 \times 3^2 \times 31\)
\(\ds \) \(=\) \(\ds 35 \, 712\)

$\blacksquare$