

A133613


Decimal digits such that for all k>=1, the number A(k) := Sum_{n = 0..k1 } a(n)*10^n satisfies the congruence 3^A(k) == A(k) (mod 10^k).


16



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
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OFFSET

0,1


COMMENTS

10adic 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 is usually defined as 3^^64 [see M. Gardner and Wikipedia], in which case only its 62 lowermost digits are guaranteed to match this sequence. To avoid such confusion, it would be best to interpret this sequence as a realvalued 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


REFERENCES

M. Gardner, Mathematical Games, Scientific American 237, 18  28 (1977).
M. RipĂ , La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, p. 1112, 6978. ISBN 9788861787896.
Ilan Vardi, "Computational Recreations in Mathematica," AddisonWesley Publishing Co., Redwood City, CA, 1991, pages 226229.


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]
Wikipedia, Graham's number
Reddit user atticdoor, Spotted an error in the comments of sequence A133613.


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

Cf. A133612, A133614, A133615, A133616, A133617, A133618, A133619, A144539, A144540, A144541, A144542, A144543, A144544.
Sequence in context: A020843 A241296 A083648 * A194622 A193010 A079082
Adjacent sequences: A133610 A133611 A133612 * A133614 A133615 A133616


KEYWORD

nonn,base


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



