The answer is 4.

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Since $7,1249$ are prime and therefore $\sigma_1(8743) = \sigma_1(7)\sigma_1(1249) = 8\times1250 = 10000$ , we see that $k \le 4$ . It is simple enough to check the values of $\sigma_1(n)$ for $1 \le n \le 1000$ to see that it is not possible for $\sigma_1(n)$ to be equal to $10$ , $100$ or $1000$ . Thus $k = \boxed{4}$ .

For interest, the other number $n$ with $\sigma_1(n) = 10000$ is $9481 = 19 \times 499$ .