A133613 Decimal digits such that for all k >= 1, the number A(k) := Sum_{n = 0..k-1} a(n)*10^n satisfies the congruence 3^A(k) == A(k) (mod 10^k).

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

Offset: 0

Author

Daniel Geisler (daniel(AT)danielgeisler.com), Dec 18 2007

Keywords

• nonn
• base

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

Examples

			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.
		

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

Crossrefs

Cf. A133612, A133614, A133615, A133616, A133617, A133618, A133619, A144539, A144540, A144541, A144542, A144543, A144544, A317905, A318478, A373387, A376842.

Programs

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 *)

Formula

a(n) = floor( A183613(n+1) / 10^n ).

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