Daniel Geisler - cp's OEIS frontend

A133618 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 8^A(k) == A(k) (mod 10^k).

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

Offset: 0

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

10-adic expansion of the iterated exponential 8^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n > 9, 8^^n == 5225856 (mod 10^7).

			658522598614530775125180015885590261391156298377201573882667037572742442437584...
8^56 == 56 (mod 100), 8^856 == 856 (mod 1000), ...

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

Robert G. Wilson v, Table of n, a(n) for n = 0..1024
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.

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

(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[8, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)

More terms from J. Luis A. Yebra, Dec 12 2008
Edited by N. J. A. Sloane, Dec 22 2008
a(68) onward from Robert G. Wilson v, Mar 06 2014

A133616 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 6^A(k) == A(k) (mod 10^k).

Original entry on oeis.org

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

Offset: 0

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

10-adic expansion of the iterated exponential 6^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n > 9, 6^^n == 7238656 (mod 10^7).

			656832744722395569767321960175060586918013794604470464024637816708501434441857...

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

Robert G. Wilson v, Table of n, a(n) for n = 0..1024
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.

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

(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[6, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)

More terms from J. Luis A. Yebra, Dec 12 2008
Edited by N. J. A. Sloane, Dec 22 2008
a(68) onward from Robert G. Wilson v, Mar 06 2014

A133615 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 5^A(k) == A(k) (mod 10^k).

Original entry on oeis.org

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

Offset: 0

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

10-adic expansion of the iterated exponential 5^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n > 9, 5^^n == 8203125 (mod 10^7).

			521302804816251394711785380951156980492298933981331774671028375173141197829625...

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

Robert G. Wilson v, Table of n, a(n) for n = 0..1024
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.

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

(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[5, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)

More terms from J. Luis A. Yebra, Dec 12 2008
Edited by N. J. A. Sloane, Dec 22 2008
a(68) onward from Robert G. Wilson v, Mar 06 2014

A133619 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 9^A(k) == A(k) (mod 10^k).

Original entry on oeis.org

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

Offset: 0

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

10-adic expansion of the iterated exponential 9^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n > 9, 9^^n == 2745289 (mod 10^7).

			982547293795780847016574305627284525700589988740419498868468199262013754161360...

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

Robert G. Wilson v, Table of n, a(n) for n = 0..1024
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.

Cf. A306686.

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

(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[9, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)

More terms from J. Luis A. Yebra, Dec 12 2008
Edited by N. J. A. Sloane, Dec 22 2008
a(68) onward from Robert G. Wilson v, Mar 06 2014

A133617 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 7^A(k) == A(k) (mod 10^k).

Original entry on oeis.org

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

Offset: 0

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

10-adic expansion of the iterated exponential 7^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n > 9, 7^^n == 5172343 (mod 10^7).

			343271565115562133346358333736086036956741826659265308652844477767549129865770...
Sequences A133612-A144544 generalize the observation that 7^343 == 343 (mod 1000).

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

Robert G. Wilson v, Table of n, a(n) for n = 0..1024
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.

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

(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[7, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)

More terms from J. Luis A. Yebra, Dec 12 2008
Edited by N. J. A. Sloane, Dec 22 2008
a(68) onward from Robert G. Wilson v, Mar 06 2014

A133614 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 4^A(k) == A(k) (mod 10^k).

Original entry on oeis.org

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

Offset: 0

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

10-adic expansion of the iterated exponential 4^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n > 9, 4^^n == 1728896 (mod 10^7).

			698827114092555203226394953143931206575634213526062954066075956906146883836488...

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

Robert G. Wilson v, Table of n, a(n) for n = 0..1024
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.

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

(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[4, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)

More terms from J. Luis A. Yebra, Dec 12 2008
Edited by N. J. A. Sloane, Dec 22 2008
a(68) onward from Robert G. Wilson v, Mar 06 2014

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).

Original entry on oeis.org

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

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

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).

			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.

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.

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)

Cf. A014221, A109405.

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

(* 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 *)

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

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).

Original entry on oeis.org

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

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

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

			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.

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.

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

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

(* 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 *)

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

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

cp's OEIS Frontend

User: Daniel Geisler

A133618 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 8^A(k) == A(k) (mod 10^k).

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A133616 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 6^A(k) == A(k) (mod 10^k).

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A133615 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 5^A(k) == A(k) (mod 10^k).

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A133619 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 9^A(k) == A(k) (mod 10^k).

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A133617 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 7^A(k) == A(k) (mod 10^k).

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A133614 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 4^A(k) == A(k) (mod 10^k).

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Mathematica

Extensions

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).

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