Return the nth digit of the sequence of aliquot series
Mathematica, 51 bytes

Array[##&amp;@@IntegerDigits[Tr@Divisors@#-#]&amp;,#][[#]]&amp;


An unnamed function which takes and returns an integer and uses 1-based indexing. Handles input 10000 instantly.

Explanation

This is a very straight-forward implementation of the definition, making use of the fact that the first n divisor sums are always enough to determine the nth digit. As usual, the reading order of golfed Mathematica is a bit funny though:

Array[...&amp;,#]...&amp;


This generates a list with all the results of applying the unnamed function on the left to all the values i from 1 to n inclusive.

...Tr@Divisors@#-#...


We start by computing the divisors of i, summing them with Tr and subtracting i itself so that it's just the sum of divisors less than i.

...IntegerDigits[...]...


This turns the result into a list of its decimal digits.

##&amp;@@...


And this removes the "list" head, so that all the digit lists are automatically concatenated in the result of Array. For more details on how ## works, see the "Sequences of arguments" section in this post.

...[[#]]


Finally, we select the nth digit from the result.