Calculate the lowest number where the sum of the sequence of numbers exceeds a given value
Jelly, 18 12 11 10 bytes

1Æs&gt;¥#ḢṄÆs


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-1 byte thanks to Mr. Xcoder!

How it works

1Æs&gt;¥#ḢṄÆs - Main link. Argument: n (integer)
1   ¥#     - Find the first n integers where...
 Æs        -   the divisor sum
   &gt;       -   is greater than the input
       Ṅ   - Print...
      Ḣ    -   the first element
        Æs - then print the divisor sum