27355 is a member of OEIS A046423: numbers requiring 3 steps to reach a prime under the prime factor concatenation procedure, here prime is 13195333. The sequence begins:

25, 35, 56, 62, 69, 72, 76, 85, 90, 100, 114, 125, 138, 143, 152, 189, 213, 221, 238, 250, 259, 260, 270, 275, 278, 303, 318, 333, 364, 366, 369, 390, 403, 441, 448, 454, 469, 470, 473, 485, 507, 512, 519, 522, 529, 559, 560, 572, 588, 591, 624, 629, 648 

There are well over 3000 of these numbers in the range up to 40000 and one way to thin them out is to impose the condition that the home prime must have at least half of its digits equal to 3. Applying this rule we end up with only 191 in the range up to 40000. The algorithm to determine these numbers is as follows (permalink):

INPUT

L=[]

for n in [2..40000]:

  if is_prime(n)==0:

    number=n

    count=0

    while is_prime(number) != 1 and count<10:

      concat=""

      F=list(factor(number))

      for f in F:

        concat+=str(f[0])*f[1]

      number=int(concat)

      count+=1

    if count==3:

      D=Integer(number).digits()

      if D.count(3)>=len(D)/2:

        L.append(n)

print(L)

print(len(L))

OUTPUT

[76, 85, 125, 143, 259, 369, 559, 591, 849, 1068, 1159, 1329, 1509, 1556, 1690, 1826, 1835, 1862, 2020, 2186, 2222, 2224, 2260, 2485, 2619, 2681, 2794, 2875, 2947, 3007, 3028, 3458, 3551, 3573, 3598, 3655, 3832, 4024, 4094, 4207, 4220, 4418, 4461, 4778, 4916, 4940, 5058, 5364, 5955, 6059, 6171, 6213, 6472, 6775, 6801, 6830, 7377, 7942, 8110, 8479, 8593, 9186, 9318, 9663, 9892, 10012, 10066, 10108, 10669, 11263, 11327, 11366, 11381, 11439, 11513, 11575, 11899, 11902, 12006, 12159, 12273, 12685, 13006, 13075, 14133, 14849, 15097, 15115, 15423, 15922, 16083, 16151, 16279, 16965, 17039, 17110, 17886, 18466, 18638, 18758, 18794, 18914, 18969, 19024, 19686, 19831, 19898, 20153, 20185, 20237, 20300, 20404, 20452, 20614, 21423, 21543, 21879, 22025, 22141, 22646, 22657, 22819, 22835, 23019, 23031, 23102, 23406, 23453, 23594, 23686, 24034, 24398, 24599, 24670, 24901, 24988, 25207, 25450, 25658, 25692, 25910, 26380, 26562, 26698, 27117, 27355, 27586, 27641, 28084, 28161, 28170, 28644, 28698, 28967, 29405, 29702, 30112, 30335, 31027, 31932, 32037, 32165, 32927, 33016, 33855, 33950, 33958, 34197, 34442, 34677, 35213, 35542, 35561, 35719, 35878, 36223, 36861, 36934, 36967, 37437, 37473, 37474, 37785, 38181, 38345, 38499, 39377, 39490, 39595, 39751, 39850]

191

The algorithm can also be applied to other digits in the range up to 40000. The results are as follows:

0 --> no numbers

1 --> 120 numbers

2 --> 2 numbers

3 --> 91 numbers

4 --> 3 numbers

5 --> 2 numbers

6 --> no numbers

7 --> 69 numbers

8 --> no numbers

9 --> 11 numbers

27355 is a Smith number, since the sum of its digits (22) coincides with the sum of the digits of its prime factors. Since it is square-free, it is also a hoax number.

27355 is a Duffinian number.

27355 stabilises after a little over 400 generations into a combination of simple still lifes and blinkers.