27568 is a member of OEIS A085844: numbers equal to a permutation of the digits of the sum of their proper divisors. The sequence can be generated as follows (permalink):

INPUT

L=[]

for n in [1..40000]:

  A=sorted(n.digits())

  total=sigma(n)-n

  B=sorted(total.digits())

  if A==B:

    L.append(n)

print(L)

OUTPUT

[6, 28, 411, 496, 604, 1305, 3664, 4086, 4672, 4896, 5046, 7785, 8128, 8739, 9331, 14535, 16012, 18342, 18585, 19648, 20634, 21534, 21628, 22365, 25911, 27568, 28108, 29160, 29188, 31185, 32091, 32271, 34956, 35898, 35925, 36172, 37698]

NOTE: Perfect numbers (A000396) are a subsequence. Perfect numbers are defined as numbers equal to the sum of their proper divisors. The numbers in this sequence are not perfect, but are "permutably perfect" since they are equal to a permutation of the sum of their proper divisors.

Note a(79)=195921 and a(80)=199251 and these are both permutations of each other. The same is true of a(90)=226944 and a(93)=244692.

27568 is a d-powerful number (and thus an energetic number) because it can be written as 2^12 + 7 + 5^6 + 6^5 + 8^2. Together with 27569, it forms a pair of energetic numbers.