Divisor Sum of 994

Example of Divisor Sum of Square-Free Integer

$\map {\sigma_1} {994} = 1728$

where $\sigma_1$ denotes the divisor sum function.

Proof

We have that:

$994 = 2 \times 7 \times 71$

Hence:

\(\ds \map {\sigma_1} {994}\)

\(=\)

\(\ds \paren {2 + 1} \paren {7 + 1} \paren {71 + 1}\)

Divisor Sum of Square-Free Integer

\(\ds \)

\(=\)

\(\ds 3 \times 8 \times 72\)

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds 2^6 \times 3^3\)

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds 1728\)

$\blacksquare$