
COMMENTS

Define a map r>A175420(r) which takes the base10 digits of r = sum_{i>=0} d_i*10^i and assigns the powertower (d_0^d_1)^d_2)^d^3... to the result. There are A055642(r)1 exponentiations in this expression. Singledigit numbers are fixed points of the map.
Starting with n, this map is iterated as often as needed to result in a singledigit number, which becomes a(n). In case the iteration does not reach a singledigit number (i.e., enters cycles with only multidigit numbers), a(n)= 1.
The entries 1 to 9 appear infinitely often in the sequence.
The entry 1 appears infinitely often in the sequence, see A175426.
After k iterations (k >= 0) we reach the following sequences:
0th step: A001477: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, ...
1st step: A175420: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 0, ...
2nd step: A175421: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 4, 9, 6, 25, 216, 6561, 4096, 1, 0, 1, 8, 49, 4096, 25, 36, 531441, 32, 22876792454961 0, 1, 6, 1, 60466176, 244140625, 101559956668416 1, 1, 1, 0, ...
3rd step: A175422: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 4, 9, 6, 25, 36, 1, 1, 1, 0, 1, 8, 6561, 1, 25, 216, 1, 8, 1, 0, 1, 6, 1, 1, 1, 1, 1, 1, 1, 0, ...
4th step: A175423: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 4, 9, 6, 25, 216, 1, 1, 1, 0, 1, 8, 1, 1, 25, 36, 1, 8, 1, 0, 1, 6, 1, 1, 1, 1, 1, 1, 1, 0, ...
