27681 is a member of OEIS A319738: numbers whose Collatz trajectories cross their initial values a record number of times, here 61 times. The records are as follows (permalink):

number   record

 1     0

 3     1

 6     4

 9     5

 14     8

 18     10

 33     13

 54     14

 97     19

 129    27

 194    28

 257    29

 294    30

 313    31

 342    32

 353    37

 398    38

 417    45

 470    46

 626    50

 9225    51

 13739   55

 14473   57

 19297   59

 27681   61

 38881   69

 112782   70

 283261   71

 427762   72

 506977   77

 831527   79

 876011   81

 1108702  82

 1168014  88

 1848673  93

 2191019  97

 5524041  99

 8546945  113

 10817230  116

 10964814  120

 11395926  128

 13506283  129

 15194569  131

 41698569  139

 46910891  141

 52774753  143

The sequence runs:

1, 3, 6, 9, 14, 18, 33, 54, 97, 129, 194, 257, 294, 313, 342, 353, 398, 417, 470, 626, 9225, 13739, 14473, 19297, 27681, 38881, ...

The confirmation that the value is 61 for 27681 is given by this algorithm: permalink.

27681 is an interprime prime because it is at equal distance from previous prime (27673) and next prime (27689).

27681 is a cyclic and D-number.

27681 can be rendered as a digit equation as follow:

2 * 7 = 6 + 8 * 1