OFFSET
0,7
COMMENTS
Any nonnegative number can be decomposed in the following way: n = a[x]b, where the brackets denote the box notation of hyperoperation.
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.
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.
LINKS
Natan Arie' Consigli, Table of n, a(n) for n = 0..260
FORMULA
EXAMPLE
a(0) = 0 because 0[+oo]n = 0.
a(1) = 0 because 1[+oo]n = 1.
a(4) = 0 because 2[+oo]2 = 4.
a(2) = 1 because 2 is prime.
a(6) = 2 because 6 is composite but not a power.
a(9) = 3 because 9 is a power but not a hyper-4 power.
a(27) = 4 because 27 is a hyper-4 power but not a hyper-5 power.
a(65536) = 5 because 65536 is a hyper-5 power but not a hyper-6 power.
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CROSSREFS
KEYWORD
nonn
AUTHOR
Natan Arie Consigli, Jul 08 2018
STATUS
approved