Divisor Sum of Non-Square Semiprime/Examples/115/Proof 2

Example of Divisor Sum of Non-Square Semiprime

$\map {\sigma_1} {115} = 144$

Proof

We have that:

$115 = 5 \times 23$

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

Hence:

\(\ds \map {\sigma_1} {115}\)

\(=\)

\(\ds \paren {5 + 1} \paren {23 + 1}\)

Divisor Sum of Non-Square Semiprime

\(\ds \)

\(=\)

\(\ds 6 \times 24\)

\(\ds \)

\(=\)

\(\ds \paren {2 \times 3} \times \paren {2^3 \times 3}\)

\(\ds \)

\(=\)

\(\ds 2^4 \times 3^2\)

\(\ds \)

\(=\)

\(\ds \paren {2^2 \times 3}^2\)

\(\ds \)

\(=\)

\(\ds 12^2\)

\(\ds \)

\(=\)

\(\ds 144\)

$\blacksquare$