Divisor Sum of 496

Example of Divisor Sum of Integer

$\map {\sigma_1} {496} = 992$

where $\sigma_1$ denotes the divisor sum function.

Proof

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:

$496 = 2^4 \times 31$

Hence:

\(\ds \map {\sigma_1} {496}\)

\(=\)

\(\ds \paren {2^5 - 1} \times \paren {31 + 1}\)

\(\ds \)

\(=\)

\(\ds 31 \times 32\)

\(\ds \)

\(=\)

\(\ds 31 \times 2^5\)

\(\ds \)

\(=\)

\(\ds 992\)

$\blacksquare$