A133612 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 2^A(k) == A(k) (mod 10^k).

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

Offset

0,1

Comments

10-adic expansion of the iterated exponential 2^^n (A014221) for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n > 9, 2^^n == 2948736 (mod 10^7).

Sequences A133612-A133619 and A144539-A144544 generalize the observation that 7^343 == 343 (mod 1000).

References

M. Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. 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..5999

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 G. Wilson v, Mathematica coding for "SuperPowerMod" and associated programs from Prof. Ian Vardi (Corrected Aug 24 2020)

Example

63784923435357051689083335895100627869682554107542682614828212121907298... - Robert G. Wilson v, Feb 22 2014
2^36 = 68719476736 == 36 (mod 100), 2^736 == 736 (mod 1000), 2^8736 == 8736 (mod 10000), etc.

Mathematica

(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[2, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Feb 22 2014 *)

Crossrefs

Cf. A014221, A109405.

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

Sequence in context: A251780 A249542 A125123 * A221149 A070392 A248268

Adjacent sequences: A133609 A133610 A133611 * A133613 A133614 A133615

Keyword

nonn,base

Author

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

Extensions

Edited by N. J. A. Sloane, Dec 22 2007 and Dec 22 2008

More terms from J. Luis A. Yebra, Dec 12 2008

a(68) onward from Robert G. Wilson v, Feb 22 2014

Status

approved