27309 is a member of OEIS A081848: number of numbers whose base-3/2 expansion (see A024629 where SageMath code be;pw can be found) has n digits, here n=25. The sequence can be generated up to 5394 after which the algorithm will time out on SageMathCell (permalink).

INPUT

def basepqExpansion(p, q, n):

  L, i = [n], 1

  while L[i-1] >= p:

    x=L[i-1]

    L[i-1]=x.mod(p)

    L.append(q*(x//p))

    i+=1

  L.reverse()

  return Integer(''.join(str(x) for x in L))

L=[basepqExpansion(3, 2, n) for n in [0..20000]]

M=[]

for n in [1..25]:

  count=0

  for x in L:

    if len(str(x))==n:

      count+=1

  M.append(count)

print(M)

OUTPUT

[3, 3, 3, 6, 9, 12, 18, 27, 42, 63, 93, 141, 210, 315, 474, 711, 1065, 1599, 2397, 3597, 5394, ... ]

Python code to generate this sequence is provided in OEIS A081848:

from itertools import islice

def A081848_gen(): # generator of terms

  yield (a:=3)

  while True:

    yield (b:=(a+1>>1)+(a&1))

    a += b

A081848_list = list(islice(A081848_gen(), 70))

27309 is a Duffinian, lucky and D-number.