%I #28 Jan 31 2024 03:58:50
%S 2,0,0,3,5,2,9,9,3,0,4,0,6,8,4,6,4,6,4,9,7,9,0,7,2,3,5,1,5,6,0,2,5,5,
%T 7,5,0,4,4,7,8,2,5,4,7,5,5,6,9,7,5,1,4,1,9,2,6,5,0,1,6,9,7,3,7,1,0,8,
%U 9,4,0,5,9,5,5,6,3,1,1,4,5,3,0,8,9,5,0
%N Decimal expansion of 2^(2^(2^(2^2))) = 2^^5.
%C 2^0 = 1, 2^1 = 2, 2^2 = 4, 2^2^2 = 2^^3 = (2^2)^2 = 16,
%C 2^2^2^2 = 2^^4 = (((2^2)^2)^2)^2 = 65536
%C so that 2^2^2^2^2 = 2^^5 = 2^(2^(2^(2^2))) = 2^65536 = 20035299304068464649790723515602557504478254755697514192650169737108940595563...
%H Chai Wah Wu, <a href="/A361100/b361100.txt">Table of n, a(n) for n = 19730..39458</a>
%H Googology Wiki, <a href="https://googology.fandom.com/wiki/Tetration">Tetration</a>.
%H Pointless Large numbers stuff by Cookiefonster, <a href="https://sites.google.com/site/pointlesslargenumberstuff/home/2/uparrows">2.02 Knuth's Up-Arrows and the Hyper-Operators</a>.
%H Robert P. Munafo, <a href="http://www.mrob.com/pub/math/seq-a094358.html">Sequence A094358, 2^^N = 1 mod N</a>.
%F Equals 2^2^2^2^2 = 2^^5 = (((((((((((((((2^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2.
%e 2003529930406846464979072351560255750447825475569751419265016973710894059556311453089506130880933348
%e (...19529 digits omitted...)
%e 5775699146577530041384717124577965048175856395072895337539755822087777506072339445587895905719156736.
%e The above example line shows the first one hundred decimal digits and the last one hundred digits with the number of unrepresented digits in parentheses.
%t nbrdgt = 100; f[base_, exp_] := RealDigits[ 10^FractionalPart[ N[ exp*Log10[ base], nbrdgt + Floor[ Log10[ exp]] + 2]], 10, nbrdgt][[1]]; f[2, 2^2^2^2]
%t IntegerDigits[2^65536][[;;100]] (* _Paolo Xausa_, Jan 31 2024 *)
%o (Python)
%o def A361100(n): return (1<<(1<<(1<<(1<<(1<<1)))))//10**(39458-n)%10 # _Chai Wah Wu_, Apr 03 2023
%Y Cf. A085667, A014221, A241291, A241292, A241293, A241294, A241295, A241296, A241297, A241298, A241299, A243913.
%K nonn,cons,easy,fini,full
%O 19730,1
%A _Marco RipĂ _, Mar 03 2023