1, 10, 100, 1000, 10000, 100000

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 $9481 = 19 \times 499$ .

Using Mathematica we can find the first few n numbers

Rest@Select[Range@100000, IntegerQ[Log10@DivisorSigma[1, #]] &]

which returns
{8743, 9481, 71193, 94981, 98671, 99301}

so the k numbers we are looking for are
4, 4, 5, 5, 5, 5

so the smallest k is 4