Divisor Sum of 472

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Example of Divisor Sum of Integer

$\map {\sigma_1} {472} = 900$

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:

$472 = 2^3 \times 59$


Hence:

\(\ds \map {\sigma_1} {472}\) \(=\) \(\ds \frac {2^4 - 1} {2 - 1} \times \frac {59^2 - 1} {59 - 1}\)
\(\ds \) \(=\) \(\ds \frac {15} 1 \times \frac {60 \times 58} {58}\) Difference of Two Squares
\(\ds \) \(=\) \(\ds 15 \times 60\)
\(\ds \) \(=\) \(\ds \paren {3 \times 5} \times \paren {2^2 \times 3 \times 5}\)
\(\ds \) \(=\) \(\ds 2^2 \times 3^2 \times 5^2\)
\(\ds \) \(=\) \(\ds \paren {2 \times 3 \times 5}^2\)
\(\ds \) \(=\) \(\ds 30^2\)
\(\ds \) \(=\) \(\ds 900\)

$\blacksquare$