Divisor Sum of 75

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Example of Divisor Sum of Integer

$\map {\sigma_1} {75} = 124$

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:

$75 = 3 \times 5^2$


Hence:

\(\ds \map {\sigma_1} {75}\) \(=\) \(\ds \frac {3^2 - 1} {3 - 1} \times \frac {5^3 - 1} {5 - 1}\)
\(\ds \) \(=\) \(\ds \frac 8 2 \times \frac {124} 4\)
\(\ds \) \(=\) \(\ds 4 \times 31\)
\(\ds \) \(=\) \(\ds 124\)

$\blacksquare$