Number Base Permutations

There are some numbers that have the same digits but in a different order when converted to another number base. Take 10144 as an example:$$10144_{10}= 41401_7$$The five digit numbers from 10000 to 40000 with this property are as follows (permalink):

10144, 10235, 10342, 10453, 10542, 11425, 11750, 12415, 12450, 12564, 12651, 13045, 13245, 13260, 13402, 13534, 13620, 14610, 15226, 15643, 15680, 16121, 16255, 16273, 16546, 16633, 18291, 19463, 19730, 21322, 21753, 21763, 21835, 23146, 23568, 24871, 25061, 25169, 26804, 26931, 29103, 29610, 30189, 30576, 31112, 31457, 31481, 32321, 32348, 34179, 34582, 35001, 35024, 35081, 35228, 35731, 36417, 37465, 38276

The table below shows the full details for the above numbers. As can be seen, they are not that frequent in the given range.

Results for range: 10000 to 40000
Decimal Number     | Base b Representation  | Base
-------------------------------------------------------
10144              | 41401                  | 7
10235              | 15032                  | 9
10342              | 42103                  | 7
10453              | 15304                  | 9
10542              | 42510                  | 7
11425              | 45211                  | 7
11750              | 17105                  | 9
12415              | 51124                  | 7
12450              | 51204                  | 7
12564              | 51426                  | 7
12651              | 51612                  | 7
13045              | 53014                  | 7
13245              | 53421                  | 7
13260              | 20163                  | 9
13402              | 20341                  | 9
13534              | 54313                  | 7
13620              | 20613                  | 9
14610              | 60411                  | 7
15226              | 62251                  | 7
15643              | 63415                  | 7
15680              | 10865                  | 11
16121              | 11126                  | 11
16255              | 65251                  | 7
16273              | 37621                  | 8
16546              | 66145                  | 7
16633              | 66331                  | 7
18291              | 12819                  | 11
19463              | 13694                  | 11
19730              | 13907                  | 11
21322              | 32221                  | 9
21753              | 52371                  | 8
21763              | 32761                  | 9
21835              | 32851                  | 9
23146              | 16432                  | 11
23568              | 35286                  | 9
24871              | 12487                  | 12
25061              | 12605                  | 12
25169              | 12695                  | 12
26804              | 40682                  | 9
26931              | 19263                  | 11
29103              | 10329                  | 13
29610              | 10629                  | 13
30189              | 10983                  | 13
30576              | 73560                  | 8
31112              | 11213                  | 13
31457              | 75341                  | 8
31481              | 11438                  | 13
32321              | 22313                  | 11
32348              | 48332                  | 9
34179              | 17943                  | 12
34582              | 52384                  | 9
35001              | 53010                  | 9
35024              | 24350                  | 11
35081              | 53108                  | 9
35228              | 53282                  | 9
35731              | 13357                  | 13
36417              | 13764                  | 13
37465              | 56347                  | 9
38276              | 26837                  | 11

Note that the bases range from 7 to 13. While this sequence does not occur in the OEIS, some subsets do. For example, OEIS  A037440:

 A037440: positive numbers having the same set of digits in bases 7 and 10.