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A175419 The single-digit number obtained by iterated mapping of r (starting with n) to a power-tower of its digits, or -1 if such a single-digit number is never reached.

%I A175419 #24 May 11 2025 09:14:41
%S A175419 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,-1,-1,1,1,1,0,1,8,
%T A175419 1,1,-1,-1,1,8,1,0,1,6,1,1,1,1,1,1,1,0,1,8,1,1,1,1,1,1,1,0,1,1,1,1,1,
%U A175419 1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,1,8,1,1,1,1,1,1,1,1
%N A175419 The single-digit number obtained by iterated mapping of r (starting with n) to a power-tower of its digits, or -1 if such a single-digit number is never reached.
%C A175419 Define a map r->A175420(r) which takes the base-10 digits of r = Sum_{i>=0} d_i*10^i and assigns the power-tower ((d_0^d_1)^d_2)^d^3... to the result. There are A055642(r)-1 exponentiations in this expression. Single-digit numbers are fixed points of the map.
%C A175419 Starting with n, this map is iterated as often as needed to result in a single-digit number, which becomes a(n). In case the iteration does not reach a single-digit number (i.e., enters cycles with only multi-digit numbers), a(n)= -1.
%C A175419 The entries 1 to 9 appear infinitely often in the sequence.
%C A175419 The entry -1 appears infinitely often in the sequence, see A175426.
%C A175419 After 1 to 4 iterations we reach sequences A175420 to A175423.
%e A175419 For n = 33: a(33) = 1 because starting with 33 we reach a single-digit 1 after 4 iterations: 3^3 = 27, 7^2 = 49, 9^4 = 6561, ((1^6)^5)^6 = 1.
%e A175419 For n = 25: a(25) = -1 because starting with 25 the iteration enters a loop of 2-digit numbers: 5^2 = 25, 5^2 = 25, ...
%Y A175419 Cf. A175420, A175421, A175422, A175423, A175424, A175425, A175426, A175427.
%K A175419 sign,base
%O A175419 0,3
%A A175419 _Jaroslav Krizek_, May 09 2010