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%I #14 Jan 11 2020 15:57:47
%S 0,0,1,1,0,1,2,1,3,3,2,1,2,1,2,2,4,1,2,1,2,2,2,1,2,3,2,4,2,1,2,1,3,2,
%T 2,2,3,1,2,2,2,1,2,1,2,2,2,1,2,3,2,1,2,1,2,2,2,1,2,1,2,1,2,2,3,2,2,1,
%U 2,2,2,1,2,1,2,2,2,2,2,1,2,3,2,1,2,2,2,2,2,1,2,2,2,2,2,2,2,1,2,2,3
%N Maximal degree x for hyperoperated representation of n = a[x]b.
%C Any nonnegative number can be decomposed in the following way: n = a[x]b, where the brackets denote the box notation of hyperoperation.
%C In this sequence we take the maximal value x where the above equation is satisfied for any nonnegative a and any nonnegative nonidentity element b.
%C If n can be circulated (n = a[+oo]b, with nonidentity element b) then a(n)=0. An identity element b is a number where we would have the trivial decomposition a[x]b = a, for some x for any a. If x=1 (addition) the identity element is b=0. If x > 1 (multiplication, exponentiation, tetration, pentation, etc.) the identity element is b=1. If n is prime, a(n)=1 because there's no x > 1 such that a[x]b = n and b > 1. If n is composite but not a nontrivial power then a(n)=2, because there would be no x > 2 such that a[x]b = n and b > 1. If n is a power but not a nontrivial hyper-4 power then a(n)=3, because there would be no x > 3 such that a[x]b = n and b > 1. If n is a hyper-4 power but not a nontrivial hyper-5 power then a(n)=3, because there would be no x > 4 such that a[x]b = n and b > 1. And so on.
%H Natan Arie' Consigli, <a href="/A302555/b302555.txt">Table of n, a(n) for n = 0..260</a>
%F a(n) = 0 if n is 0, 1 or 4.
%F a(n) = 1 if n is in A000040 (the primes).
%F a(n) = 2 if n is in A106543 (non-powered composites).
%F a(n) = 3 if n is in A302554 (non-hyper-4 power powers).
%F a(n) = 4 if n is in A302553 (non-hyper-5 power hyper-4 powers).
%F ...
%e a(0) = 0 because 0[+oo]n = 0.
%e a(1) = 0 because 1[+oo]n = 1.
%e a(4) = 0 because 2[+oo]2 = 4.
%e a(2) = 1 because 2 is prime.
%e a(6) = 2 because 6 is composite but not a power.
%e a(9) = 3 because 9 is a power but not a hyper-4 power.
%e a(27) = 4 because 27 is a hyper-4 power but not a hyper-5 power.
%e a(65536) = 5 because 65536 is a hyper-5 power but not a hyper-6 power.
%e ...
%Y Cf. A000040, A106543, A302553, A302554.
%K nonn
%O 0,7
%A _Natan Arie Consigli_, Jul 08 2018
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