Sigma Function of 1210

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Example of Sigma Function of Integer

$\sigma \left({1210}\right) = 2394$

where $\sigma$ denotes the $\sigma$ function.

Proof

$\displaystyle \sigma \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$

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

We have that:

$1210 = 2 \times 5 \times 11^2$

Hence:

 $\displaystyle \sigma \left({1210}\right)$ $=$ $\displaystyle \left({2 + 1}\right) \times \left({5 + 1}\right) \times \frac {11^3 - 1} {11 - 1}$ $\displaystyle$ $=$ $\displaystyle 3 \times 6 \times \frac {1330} {10}$ $\displaystyle$ $=$ $\displaystyle 3 \times 6 \times 133$ $\displaystyle$ $=$ $\displaystyle 3 \times \left({2 \times 3}\right) \times \left({7 \times 19}\right)$ $\displaystyle$ $=$ $\displaystyle 2 \times 3^2 \times 7 \times 19$ $\displaystyle$ $=$ $\displaystyle 2394$

$\blacksquare$