OFFSET
0,1
COMMENTS
10-adic expansion of the iterated exponential 3^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n>9, 3^^n == 4195387 (mod 10^7).
This sequence also gives many final digits of Graham's number ...399618993967905496638003222348723967018485186439059104575627262464195387. - Paul Muljadi, Sep 08 2008 and J. Luis A. Yebra, Dec 22 2008
Graham's number can be represented as G(64):=3^^3^^...^^3 [see M. Gardner and Wikipedia], in which case its G(63) lowermost digits are guaranteed to match this sequence (i.e., the convergence speed of the base 3 is unitary - see A317905). To avoid such confusion, it would be best to interpret this sequence as a real-valued constant 0.783591464..., corresponding to 3^^k in the limit of k->infinity, and call it Graham's constant G(3). Generalizations to G(n) and G(n,base) are obvious. - Stanislav Sykora, Nov 07 2015
Let G(64) be Graham's number. Let b and c be two (strictly) positive integers so that the super-logarithm base b of c (i.e., slog_b(c)) is well defined. Then, this sequence gives the slog_3(G(64))-1 final digits of G(64) since the congruence speed of 3 is equal to 0 at height 1 while it is 1 for all the integer hyperexponents above 0 (i.e., 3 is characterized by a constant congruence speed of 1, as proved by Lemma 1 of "On the congruence speed of tetration" and also confirmed by Equation (16) of "Number of stable digits of any integer tetration" - see Links). On the other hand, the difference between the slog_3(G(64))-th rightmost digit of G(64) and a(slog_3(G(64))) is congruent to 6 modulo 10 (since the asymptotic phase shift of 3 is [4,6] - see A376842). - Marco Ripà, Oct 17 2024
REFERENCES
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. 11-12, 69-78. ISBN 978-88-6178-789-6.
Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 0..10039
J. Jimenez Urroz and J. Luis A. Yebra, On the equation a^x == x (mod b^n), J. Int. Seq. 12 (2009) #09.8.8.
Robert P. Munafo, Large Numbers [From Robert G. Wilson v, May 07 2010]
Reddit user atticdoor, Spotted an error in the comments of sequence A133613.
Marco Ripà, On the constant congruence speed of tetration, Notes on Number Theory and Discrete Mathematics, Volume 26, 2020, Number 3, Pages 245—260.
Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43-61.
Wikipedia, Graham's number
FORMULA
a(n) = floor( A183613(n+1) / 10^n ).
EXAMPLE
783591464262726575401950934681584810769327843222300836694509769399816993697535...
Consider the sequence 3^^n: 1, 3, 27, 7625597484987, ... From 3^^3 = 7625597484987 onwards, all terms end with the digits 87. This follows from Euler's generalization of Fermat's little theorem.
MATHEMATICA
(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[3, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)
CROSSREFS
KEYWORD
nonn,base,changed
AUTHOR
Daniel Geisler (daniel(AT)danielgeisler.com), Dec 18 2007
EXTENSIONS
More terms from J. Luis A. Yebra, Dec 12 2008
Edited by N. J. A. Sloane, Dec 22 2008
More terms from Robert G. Wilson v, May 07 2010
STATUS
approved