Numbers As Permutations Of Their Arithmetic Derivatives

A very few numbers have arithmetic derivatives that equal to themselves. These are numbers of the form \(n^n \) where \(n\) is prime. Thus we have \(4 =2^2, 27 =3^3, 3125 =5^5\) etc. What about numbers that have arithmetic derivatives with the same digits but in a different order? The number associated with my diurnal age, \( \textbf{27891}\) today is one such number because:$$ \begin{align} 27891' &= (3^3 \times 1033)' \\ &=(3^3)' \times 1033 + 3^3 \times 1033' \\ &=3^3 \times 1033 + 3^3 \\ &= 27981 + 27 \\&=27918\end{align}$$In the range up to 40000, there are 43 such numbers including those are remain the same such as 4, 27, 3125 etc. The numbers are (permalink):

4, 27, 94, 308, 526, 594, 950, 1208, 1269, 1647, 2403, 3125, 5589, 5643, 5926, 6934, 9369, 10503, 10568, 11084, 11284, 12404, 12447, 13130, 13500, 14024, 14769, 17469, 17847, 18036, 20358, 20547, 20852, 25569, 27891, 28647, 29835, 34803, 36068, 36180, 36747, 38396, 39069

Another sequence can be declared whose members are the arithmetic derivatives themselves. In other words, numbers that are the arithmetic derivatives of numbers with identical digits but in a possibly different order (permalink):

4, 27, 49, 265, 380, 905, 945, 1296, 1674, 1820, 2430, 2965, 3125, 3469, 6453, 9396, 9558, 10331, 10530, 11804, 12474, 14128, 14204, 14796, 15860, 17496, 17874, 20574, 21044, 22508, 25596, 27918, 28674, 32805, 34830, 35100, 36180, 36774, 36860, 39096, 39836, 39852, 80136

Here is a breakdown of each number's factorisation and permutation:

  number   factors              permutation

  4        2^2                  4

  27       3^3                  27

  94       2 * 47               49

  308      2^2 * 7 * 11         380

  526      2 * 263              265

  594      2 * 3^3 * 11         945

  950      2 * 5^2 * 19         905

  1208     2^3 * 151            1820

  1269     3^3 * 47             1296

  1647     3^3 * 61             1674

  2403     3^3 * 89             2430

  3125     5^5                  3125

  5589     3^5 * 23             9558

  5643     3^3 * 11 * 19        6453

  5926     2 * 2963             2965

  6934     2 * 3467             3469

  9369     3^3 * 347            9396

  10503    3^3 * 389            10530

  10568    2^3 * 1321           15860

  11084    2^2 * 17 * 163       11804

  11284    2^2 * 7 * 13 * 31    14128

  12404    2^2 * 7 * 443        14204

  12447    3^3 * 461            12474

  13130    2 * 5 * 13 * 101     10331

  13500    2^2 * 3^3 * 5^3      35100

  14024    2^3 * 1753           21044

  14769    3^3 * 547            14796

  17469    3^3 * 647            17496

  17847    3^3 * 661            17874

  18036    2^2 * 3^3 * 167      36180

  20358    2 * 3^3 * 13 * 29    32805

  20547    3^3 * 761            20574

  20852    2^2 * 13 * 401       22508

  25569    3^3 * 947            25596

  27891    3^3 * 1033           27918

  28647    3^3 * 1061           28674

  29835    3^3 * 5 * 13 * 17    39852

  34803    3^3 * 1289           34830

  36068    2^2 * 71 * 127       36860

  36180    2^2 * 3^3 * 5 * 67   80136

  36747    3^3 * 1361           36774

  38396    2^2 * 29 * 331       39836

  39069    3^3 * 1447           39096

These numbers constitute OEIS A225902. Some of these numbers are also ABCD numbers and these are:

1269, 10503, 12447, 14769, 17469, 17847, 18036, 20358, 20547, 27891, 28647, 29835

For example, 17892 + 27198 + \( \textbf{27891} \) = 72981