Divisor Sum of 1050

Example of Divisor Sum of Integer

$\map {\sigma_1} {1050} = 2976$

where $\sigma_1$ denotes the divisor sum function.

Proof

From Divisor Sum of Integer: Corollary

$\ds \map {\sigma_1} n = \prod_{\substack {1 \mathop \le i \mathop \le r \\ k_i \mathop > 1} } \frac {p_i^{k_i + 1} - 1} {p_i - 1} \prod_{\substack {1 \mathop \le i \mathop \le r \\ k_i \mathop = 1} } \paren {p_i + 1}$

where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.

We have that:

$1050 = 2 \times 3 \times 5^2 \times 7$

Hence:

\(\ds \map {\sigma_1} {1050}\)

\(=\)

\(\ds \paren {2 + 1} \paren {3 + 1} \times \frac {5^3 - 1} {5 - 1} \times \paren {7 + 1}\)

\(\ds \)

\(=\)

\(\ds 3 \times 4 \times \frac {124} 4 \times 8\)

\(\ds \)

\(=\)

\(\ds 3 \times 2^2 \times 31 \times 2^3\)

\(\ds \)

\(=\)

\(\ds 2^5 \times 3 \times 31\)

\(\ds \)

\(=\)

\(\ds 2976\)

$\blacksquare$