Divisor Sum of Power of Prime/Examples/81

Example of Divisor Sum of Power of Prime

$\map {\sigma_1} {81} = 121$

where $\sigma_1$ denotes the divisor sum function.

Proof

From Divisor Sum of Power of Prime:

$\map {\sigma_1} {p^k} = \dfrac {p^{k + 1} - 1} {p_i - 1}$

We have that:

$81 = 3^4$

Hence:

\(\ds \map {\sigma_1} {81}\)

\(=\)

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

\(\ds \)

\(=\)

\(\ds \frac {242} 2\)

\(\ds \)

\(=\)

\(\ds 121\)

\(\ds \)

\(=\)

\(\ds 11^2\)

$\blacksquare$