27540 is a product of a power of 2, a 4k+3 prime raised to an even power and two 4k+1 primes. Therefore it can be expressed as a sum of two squares in two ways viz. 36^2+162^2 and 108^2+126^2.

It can be noted that 27540 is a also member of OEIS A338485: primitive numbers that are the sum of the squares of two of their distinct divisors. Here 36^2+162^2=27540. To generate a list of such numbers, the following algorithm can be used (permalink):

INPUT

L=[]

for number in [1..40000]:

  D=divisors(number)

  if len(D)>=2:

    C=Combinations(D,2)

    for c in C:

      if c[0]^2+c[1]^2==number:

        L.append(number)

print(L)

print(len(L))

P=[]

for n in L:

  upper=int(sqrt(n))

  OK=1

  for k in [2..upper]:

    if n/k^2 in L:

      OK=0

      break

  if OK==1:

    P.append(n)

print(P)

print(len(P))

OUTPUT

[20, 80, 90, 180, 272, 320, 360, 468, 500, 650, 720, 810, 980, 1088, 1280, 1332, 1440, 1620, 1872, 2000, 2250, 2420, 2448, 2450, 2600, 2880, 2900, 3240, 3380, 3600, 3920, 4160, 4212, 4352, 4410, 4500, 5120, 5328, 5760, 5780, 5850, 6480, 6642, 6800, 7220, 7290, 7488, 7650, 8000, 8820, 9000, 9680, 9792, 9800, 10100, 10388, 10400, 10580, 10890, 11520, 11600, 11700, 11988, 12500, 12960, 13328, 13520, 14400, 14580, 14762, 15210, 15680, 16250, 16400, 16640, 16820, 16848, 17408, 17640, 18000, 19220, 20250, 20480, 20880, 21312, 21780, 22032, 22050, 22932, 23040, 23120, 23400, 24500, 25578, 25920, 26010, 26100, 26568, 27200, 27380, 27540, 28730, 28880, 29160, 29952, 30420, 30600, 31850, 32000, 32400, 32490, 32912, 33300, 33620, 35280, 36000, 36980, 37440, 37908, 38612, 38720, 39168, 39200, 39690]

124

[20, 90, 272, 468, 650, 1332, 2450, 2900, 3600, 4160, 6642, 7650, 10100, 10388, 14762, 16400, 20880, 25578, 27540, 28730, 38612]

21

27540 is a member of OEIS A063663: numbers which can be written as b^2*c^2*(b^2+c^2), here b=4 and c=81 or vice versa. The sequence can be generated as follows:

INPUT

L=[]

S=[n^2 for n in [0..100]]

for a in S:

  for b in S:

    number=a*b*(a+b)

    L.append(number)

    if number==27540 and a<b:

      print(number,a,b)

print()

print(sorted(Set(L)))

OUTPUT

27540 4 81

[0, 2, 20, 90, 128, 272, 468, 650, 1280, 1332, 1458, 2450, 2900, 3600, 4160, 5760, 6642, 7650, 8192, 10100, 10388, 14580, 14762, 16400, 17408, 20880, 25578, 27540, 28730, 29952, 31250, 38612]

27540 is an interprime number between twin primes with 60 divisors. The only such numbers in the range up to 40000 with this property are 12240, 13680, 21060, 23040, 27540, 28350, 31248 and 37200.

27540 is a happy, interprime, tau, hoax, Harshad, practical, abundant, Zumkeller and pseudoperfect number.

27540 can be rendered as a digit equation as follows:

2 + 7 - 5 - 4 = 0