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%I #38 Feb 21 2023 13:23:10
%S 0,1,1,7,1,1,2,3,6,4,1,3,4,6,1,3,1,3,1,1,3,2,3,5,5,2,2,2,8,1,1,9,1,2,
%T 3,4,8,2,4,1,1,2,8,3,2,1,4,2,5,1,6,7,2,2,2,2,2,2,8,4,1,4,8,1,5,8,4,1,
%U 4,1,2,1,9,6,6,2,1,1,7,6,1,7,7,2,4,1,8,6,1,7,1,1,3,1,2,6,3,5,1,1,1,2,2,5,4
%N Initial digit of the decimal expansion of n^(n^n) or n^^3 (in _Don Knuth_'s up-arrow notation).
%C 0^^3 = 0 since 0^^k = 1 for even k, 0 for odd k, k >= 0.
%C Conjecture: the distribution of the initial digits obey Zipf's law.
%C The distribution of the first 1000 terms beginning with 1: 302, 196, 124, 91, 72, 46, 71, 53, 45.
%H Robert P. Munafo and Robert G. Wilson v, <a href="/A241299/b241299.txt">Table of n, a(n) for n = 0..1000</a>
%H Cut the Knot.org, <a href="http://www.cut-the-knot.org/do_you_know/zipfLaw.shtml">Benford's Law and Zipf's Law</a>, A. Bogomolny, Zipf's Law, Benford's Law from Interactive Mathematics Miscellany and Puzzles.
%H Hans Havermann, <a href="http://chesswanks.com/seq/n%5en%5en/">Next 5 terms</a>.
%H M. E. J. Newman, <a href="http://arxiv.org/abs/cond-mat/0412004">Power laws, Pareto distributions and Zipf's law</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JoyceSequence.html">Joyce Sequence</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Knuth's_up-arrow_notation">Knuth's up-arrow notation</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Zipf's_law">Zipf's law</a>.
%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>.
%F For n > 0, a(n) = floor(t/10^floor(log_10(t))) where t = n^(n^n).
%F a(n) = A000030(A002488). - _Omar E. Pol_, Jul 04 2019
%e a(0) = 0, a(1) = 1, a(2) = 1 because 2^(2^2) = 16, a(3) = 7 because 3^(3^3) = 7625597484987 and its initial digit is 7, etc.
%t g[n_] := Quotient[n^p, 10^(Floor[ p*Log10@ n] - (1004 + p))]; f[n_] := Block[{p = n}, Quotient[ Nest[ g@ # &, p, p], 10^(1004 + p)]]; Array[f, 105, 0]
%Y Cf. A000030, A000312, A002488, A066022, A241291, A241292, A241293, A241294, A241295, A241296, A241297, A241298.
%K nonn,base,easy
%O 0,4
%A _Robert Munafo_ and _Robert G. Wilson v_, Apr 18 2014
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