A097374 Perfect 4-composites: a perfect 4-composite is a natural number that can be represented in the form a^(a^(a^........(a^(a) ) ) ) for some natural number a and some number b>=1 of up-arrows.

4, 16, 27, 256, 3125, 46656, 65536, 823543, 16777216, 387420489, 10000000000, 285311670611, 7625597484987, 8916100448256, 302875106592253, 11112006825558016, 437893890380859375, 18446744073709551616, 827240261886336764177, 39346408075296537575424, 1978419655660313589123979, 104857600000000000000000000

Offset: 1

Author

Ashutosh (ashu(AT)iitk.ac.in), Sep 18 2004

Keywords

• nonn

Comments

From Natan Arie Consigli, Jan 17 2016: (Start)

Also, natural numbers of the form H_4(a,b) with a,b > 1. See A054871 for definitions and key links.

Let a and b be positive. a is a unit if there exist b such that a*b=1. The only unit is 1 because only 1*1=1.

x = a*b is composite (in hyper-2) if a,b are nonunits.

In hyper-4 context the only unit is 1 since a[4]b = 1 if and only if a=1.

Hyper 4-composites are numbers of the form H_4(a,b) where a,b are nonunits. This is why for 4-composites we have a,b > 1.

1 and 0 are non-4-composites since H_4(a,b) > 1 if a,b are positive nonunits. (End)

Examples

			4-composites include:
H_4(5,2)= 5^5 = 3125;
H_4(3,3) = 3^3^3 = 3^27 = 7625597484987;
H_4(2,4) = 2^2^2^2 = 2^2^4 = 2^16 = 65536;
		

Crossrefs

Cf. A257309 (nontrivial hyper-4 powers H_4(a,b) with b<>1).

Programs

Magma

[4,16] cat [n^n: n in [3..20]]; // Vincenzo Librandi, Jan 18 2016

Mathematica

Join[{4, 16}, Table[n^n, {n, 3, 20}]] (* Vincenzo Librandi, Jan 18 2016 *)

Formula

a(n) = A257309(n+2).

Extensions

Corrected by Natan Arie Consigli, Jan 17 2016