Divisor Sum of Non-Square Semiprime/Examples/94

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Example of Divisor Sum of Non-Square Semiprime

$\map {\sigma_1} {94} = 144$

where $\sigma_1$ denotes the divisor sum function.


Proof 1

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:

$94 = 2 \times 47$


Hence:

\(\ds \map {\sigma_1} {94}\) \(=\) \(\ds \frac {2^2 - 1} {2 - 1} \times \frac {47^2 - 1} {47 - 1}\)
\(\ds \) \(=\) \(\ds \frac 3 1 \times \frac {46 \times 48} {46}\) Difference of Two Squares
\(\ds \) \(=\) \(\ds 3 \times 48\)
\(\ds \) \(=\) \(\ds 3 \times \paren {2^4 \times 3}\)
\(\ds \) \(=\) \(\ds 2^4 \times 3^2\)
\(\ds \) \(=\) \(\ds \paren {2^2 \times 3}^2\)
\(\ds \) \(=\) \(\ds 12^2\)
\(\ds \) \(=\) \(\ds 144\)

$\blacksquare$


Proof 2

We have that:

$94 = 2 \times 47$

and so by definition is a semiprime whose prime factors are distinct.


Hence:

\(\ds \map {\sigma_1} {94}\) \(=\) \(\ds \paren {2 + 1} \paren {47 + 1}\) Divisor Sum of Non-Square Semiprime
\(\ds \) \(=\) \(\ds 3 \times 48\)
\(\ds \) \(=\) \(\ds 3 \times \paren {2^4 \times 3}\)
\(\ds \) \(=\) \(\ds 2^4 \times 3^2\)
\(\ds \) \(=\) \(\ds \paren {2^2 \times 3}^2\)
\(\ds \) \(=\) \(\ds 12^2\)
\(\ds \) \(=\) \(\ds 144\)

$\blacksquare$