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Robert P. Munafo, <a href="http://www.mrob.com/pub/math/largenum-4.html#graham">Large Numbers</a> [From _Robert G. Wilson v_, May 07 2010]>

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Marco Ripà: I see... I think the name would be removed. I took the whole reference, including Robert's signature from another Cf. sequence.
I'll remove the signature, is it ok?

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Michel Marcus: [From Robert G. Wilson v, May 07 2010] is weird considering 	
Marco Ripà, Aug 26 2018

Marco Ripà: Sorry? I'm not sure I've understood the comment... what does it mean "considering" me?
I started to study tetration at the beginning of 2010, but the whole book was published in 2011. It is focused on pseudo-congruence (analyzing the figures on the right of the frozen digits and so on... it is related to chaos theory, partitions, Charmichael's lambda function and so on).
I'm back on this topic since I started to talk about recreational mathematics on my YouTube channel since a couple of weeks ago.

Michel Marcus: I don't understand how Robert could add a reference in May 2010 to a sequence that has been created in August 2018

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

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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>=89, 1984^^n == 98703616 ((mod 10^8).) == 98703616.

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.

allocatedDecimal digits such that for all k>=1, the number A(k) := Sum_{n = 0..k-1 } a(n)*10^n satisfies the congruence Marco1984^A(k) == A(k) (mod Ripà10^k).

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

1,1

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>=8, 1984^^n == 98703616 (mod 10^8).

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.

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

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

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

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.

Robert P. Munafo, <a href="http://www.mrob.com/pub/math/largenum-4.html#graham">Large Numbers</a> [From Robert G. Wilson v, May 07 2010]

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

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

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

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

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.

Cf. A133612, A133613, A133614, A133615, A133616, A133617, A133618, A133619, A144539, A144540, A144541, A144542, A144543, A144544.

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Marco Ripà, Aug 26 2018

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