Jelly, 7 bytes
1Æṣ=$#Ṫ6Æṣ=$#Ṫ
How it works
6Æṣ=$#Ṫ Main link. Argument: n
6 Set the return value to 6.
# Execute the link to the left with argument k = 6, 7, 8, ... until n
values of k result in a truthy value. Yield the array of matches.
$ Combine the two links to the left into a monadic chain.
Æṣ Compute the sum of k's proper divisors.
= Compare the result with k.
Ṫ Tail; extract the last match.
Jelly, 7 bytes
6Æṣ=$#Ṫ
How it works
6Æṣ=$#Ṫ Main link. Argument: n
6 Set the return value to 6.
# Execute the link to the left with argument k = 6, 7, 8, ... until n
values of k result in a truthy value. Yield the array of matches.
$ Combine the two links to the left into a monadic chain.
Æṣ Compute the sum of k's proper divisors.
= Compare the result with k.
Ṫ Tail; extract the last match.