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%I #28 Sep 08 2022 08:45:14
%S 4,16,27,256,3125,46656,65536,823543,16777216,387420489,10000000000,
%T 285311670611,7625597484987,8916100448256,302875106592253,
%U 11112006825558016,437893890380859375,18446744073709551616,827240261886336764177,39346408075296537575424,1978419655660313589123979,104857600000000000000000000
%N 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.
%C From _Natan Arie Consigli_, Jan 17 2016: (Start)
%C Also, natural numbers of the form H_4(a,b) with a,b > 1. See A054871 for definitions and key links.
%C 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.
%C x = a*b is composite (in hyper-2) if a,b are nonunits.
%C In hyper-4 context the only unit is 1 since a[4]b = 1 if and only if a=1.
%C 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.
%C 1 and 0 are non-4-composites since H_4(a,b) > 1 if a,b are positive nonunits. (End)
%F a(n) = A257309(n+2).
%e 4-composites include:
%e H_4(5,2)= 5^5 = 3125;
%e H_4(3,3) = 3^3^3 = 3^27 = 7625597484987;
%e H_4(2,4) = 2^2^2^2 = 2^2^4 = 2^16 = 65536;
%t Join[{4, 16}, Table[n^n, {n, 3, 20}]] (* _Vincenzo Librandi_, Jan 18 2016 *)
%o (Magma) [4,16] cat [n^n: n in [3..20]]; // _Vincenzo Librandi_, Jan 18 2016
%Y Cf. A257309 (nontrivial hyper-4 powers H_4(a,b) with b<>1).
%K nonn
%O 1,1
%A Ashutosh (ashu(AT)iitk.ac.in), Sep 18 2004
%E Corrected by _Natan Arie Consigli_, Jan 17 2016
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