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A302555 Maximal degree x for hyperoperated representation of n = a[x]b. 1
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
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

a(n) = 0 if n is 0, 1 or 4.

a(n) = 1 if n is in A000040 (the primes).

a(n) = 2 if n is in A106543 (non-powered composites).

a(n) = 3 if n is in A302554 (non-hyper-4 power powers).

a(n) = 4 if n is in A302553 (non-hyper-5 power hyper-4 powers).

...

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.

...

CROSSREFS

Cf. A000040, A106543, A302553, A302554.

Sequence in context: A088074 A071463 A308530 * A262209 A324338 A047679

Adjacent sequences:  A302552 A302553 A302554 * A302556 A302557 A302558

KEYWORD

nonn

AUTHOR

Natan Arie' Consigli, Jul 08 2018

STATUS

approved

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Last modified September 21 23:55 EDT 2019. Contains 327286 sequences. (Running on oeis4.)