27369 is a product of a 4k+3 prime (3) raised to an even power (2) and a 4k+1 prime (3041). Thus it can be expressed as a sum of two squares in one way only viz. 165^2 + 12^2.

27369 is a member of OEIS A246421: numbers n such that (n + digit sum of n) and (n + digit product of n) are nontrivial permutations of the digits of n. See blog post titled Permutations Involving Sum and Product of Digits. The sequence can be generated as follows (permalink):

INPUT

L=[]

upper=40000

for n in [1..upper]:

  if 0 not in n.digits():

    S=n+sum(n.digits())

    P=n+prod(n.digits())

    if sorted(n.digits())==sorted(S.digits()) and sorted(n.digits())==sorted(P.digits()):

      L.append(n)

      print(n,S,P)

print()

print(L)

print("There are",len(L),"such numbers in the range up to",upper)

OUTPUT

5769 5796 7659

14346 14364 14634

27369 27396 29637

[5769, 14346, 27369]

There are 3 such numbers in the range up to 40000

27369 is a d-powerful number because it can be written as 2^11 + 7^5 + 3^6 + 6^5 + 9.

27369 stabilises after about 117 generations to a block, a beehive and a blinker.