Divisor Sum of Non-Square Semiprime/Examples/15

Example of Divisor Sum of Non-Square Semiprime

$\map {\sigma_1} {15} = 24$

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:

$15 = 3 \times 5$

Hence:

\(\ds \map {\sigma_1} {15}\)

\(=\)

\(\ds \frac {3^2 - 1} {3 - 1} \times \frac {5^2 - 1} {5 - 1}\)

\(\ds \)

\(=\)

\(\ds \frac 8 2 \times \frac {24} 4\)

\(\ds \)

\(=\)

\(\ds 4 \times 6\)

\(\ds \)

\(=\)

\(\ds 24\)

$\blacksquare$

Proof 2

We have that:

$15 = 3 \times 5$

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

Hence:

\(\ds \map {\sigma_1} {15}\)

\(=\)

\(\ds \paren {3 + 1} \paren {5 + 1}\)

Divisor Sum of Non-Square Semiprime

\(\ds \)

\(=\)

\(\ds 4 \times 6\)

\(\ds \)

\(=\)

\(\ds 24\)

$\blacksquare$