Divisor Sum of 217

Example of Divisor Sum of Non-Square Semiprime

$\map {\sigma_1} {217} = 256$

where $\sigma_1$ denotes the divisor sum function.

Proof

We have that:

$217 = 7 \times 31$

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

Hence:

\(\ds \map {\sigma_1} {217}\)

\(=\)

\(\ds \paren {7 + 1} \paren {31 + 1}\)

Divisor Sum of Non-Square Semiprime

\(\ds \)

\(=\)

\(\ds 8 \times 32\)

\(\ds \)

\(=\)

\(\ds 2^3 \times 2^5\)

\(\ds \)

\(=\)

\(\ds 2^8\)

\(\ds \)

\(=\)

\(\ds \paren {2^4}^2\)

\(\ds \)

\(=\)

\(\ds 16^2\)

\(\ds \)

\(=\)

\(\ds 256\)

$\blacksquare$