%I #10 Sep 23 2018 23:14:03

%S 6,1,6,3,0,7,8,9,3,0,7,1,4,5,9,1,2,0,3,2,9,4,8,4,0,0,1,0,9,0,4,5,1,0,

%T 2,3,9,2,0,5,0,9,4,2,6,9,0,5,3,3,8,6,2,2,8,4,6,3,8,5,1,9,2,3,7,7,8,9,

%U 0,0,2,8,3,9,2,7,0,0,1,0,7,4,9,0,3,3,5

%N Decimal digits such that for all k>=1, the number A(k) := Sum_{n = 0..k-1 } a(n)*10^n satisfies the congruence 1984^A(k) == A(k) (mod 10^k).

%C 10-adic expansion of the iterated exponential 1984^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n>=9, 1984^^n(mod 10^8) == 98703616.

%C 1984^^n, for any n>=188, appears in M. Ripà's book "La strana coda della serie n^n^...^n", where the author took his birth year (1984), as a random base in order to prove some general properties about tetration, and calculating 1984^^n(mod 10^187) as a test for his paper-and-pencil procedure.

%D M. Gardner, Mathematical Games, Scientific American 237, 18 - 28 (1977).

%D M. Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, p. 78-79. ISBN 978-88-6178-789-6.

%D Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.

%H J. Jimenez Urroz and J. Luis A. Yebra, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Yebra/yebra4.html">On the equation a^x == x (mod b^n)</a>, J. Int. Seq. 12 (2009) #09.8.8.

%H Robert P. Munafo, <a href="http://www.mrob.com/pub/math/largenum-4.html#graham">Large Numbers</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Graham&#39;s_number">Graham's number</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>

%e 1984^^1984 (mod 10^8) == 98703616.

%e Thus, 1984^^1984 = ...61630789307145912032948400109045102(...)7490335.

%e Consider the sequence 1984^^n: 1984, 1984^1984, 1984^(1984^1984), ... From 1984^^3 onwards, all terms end with the digits 16. This follows from Euler's generalization of Fermat's little theorem.

%Y Cf. A133612, A133613, A133614, A133615, A133616, A133617, A133618, A133619, A144539, A144540, A144541, A144542, A144543, A144544, A317824, A317903, A317905.

%K nonn,base

%O 1,1

%A _Marco Ripà_, Aug 26 2018