Return the nth digit of the sequence of aliquot series

Mathematica, 51 bytes

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

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[...&,#]...&

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.

##&@@...

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.

05AB1E, 14 11 10 bytes

Compute n = 9999 in about 15 seconds. Code:

ÌL€Ñ€¨OJ¹è

Explanation:

Ì           # Increment implicit input by 2
 L          # Generate the list [1 .. input + 2]
  ۄ        # For each, get the divisors
    ۬      # For each, pop the last one out
      O     # Sum all the arrays in the array
       J    # Join them all together
        ¹è  # Get the nth element

Uses the CP-1252 encoding. Try it online!.

Pyth, 26 21 20 15 bytes

@sm`sf!%dTtUdSh

Try it online. Test suite.

Uses 0-based indexing. The program is O(n²) and completes for n = 9999 in about 14 minutes on my 2008 machine.