%I #27 Feb 10 2022 22:53:07
%S 5,2,1,3,0,2,8,0,4,8,1,6,2,5,1,3,9,4,7,1,1,7,8,5,3,8,0,9,5,1,1,5,6,9,
%T 8,0,4,9,2,2,9,8,9,3,3,9,8,1,3,3,1,7,7,4,6,7,1,0,2,8,3,7,5,1,7,3,1,4,
%U 1,1,9,7,8,2,9,6,2,5,5,5,3,3,0,9,0,4,7,3,1,8,5,7,4,6,9,7,2,3,0,8,9,2,6,1,4
%N Unique sequence of digits a(0), a(1), a(2), ... such that for all k >= 2, the number A(k) := Sum_{n = 0..k-1} a(n)*10^n satisfies 5^A(k) == A(k) (mod 10^k).
%C 10-adic expansion of the iterated exponential 5^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n > 9, 5^^n == 8203125 (mod 10^7).
%D M. RipĂ , La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, p. 69-78. 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 Robert G. Wilson v, <a href="/A133615/b133615.txt">Table of n, a(n) for n = 0..1024</a>
%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.
%e 521302804816251394711785380951156980492298933981331774671028375173141197829625...
%t (* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[5, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* _Robert G. Wilson v_, Mar 06 2014 *)
%Y Cf. A133612, A133613, A133614, A133616, A133617, A133618, A133619, A144539, A144540, A144541, A144542, A144543, A144544.
%K nonn,base
%O 0,1
%A Daniel Geisler (daniel(AT)danielgeisler.com), Dec 18 2007
%E More terms from J. Luis A. Yebra, Dec 12 2008
%E Edited by _N. J. A. Sloane_, Dec 22 2008
%E a(68) onward from _Robert G. Wilson v_, Mar 06 2014