A109405 -id:A109405 - cp's OEIS frontend

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

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

A064541 Numbers k such that 2^k ends in k.

Original entry on oeis.org

36, 736, 8736, 48736, 948736, 2948736, 32948736, 432948736, 3432948736, 53432948736, 353432948736, 5353432948736, 75353432948736, 5075353432948736, 15075353432948736, 615075353432948736, 8615075353432948736, 98615075353432948736, 8098615075353432948736, 38098615075353432948736

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 08 2001

There is no term with 15 digits.

			2^36 = 68719476736 which ends in 36.

Index entries for sequences related to final digits of numbers

Cf. A121319, A109405, A206636.
The leading digits are listed in A064540.
Digits read backwards form A133612.

a[1] = 36; a[n_] := a[n] = For[ida = IntegerDigits[a[n-1]]; k = 1, True, k++, idk = IntegerDigits[k]; pm = PowerMod[2, an = FromDigits[Join[idk, ida]], 10^IntegerLength[an]]; If[pm == an, Return[an]]]; Array[a, 20] (* Jean-François Alcover, Feb 15 2018 *)

a(n+1) is a suffix of 2^a(n) formed by a nonzero digit followed by a number of zeros and a(n). E.g., a(13)=75353432948736 and 2^a(13) ends with ...15075353432948736, hence a(14)=5075353432948736. - Max Alekseyev, Apr 18 2007

Can be obtained from A109405 by removing all repeats. - Max Alekseyev, May 11 2007

A113627 a(n) is the smallest number k such that k and 2^k have the same last n digits. Here k may have fewer than n digits and can be padded with leading zeros (cf. A121319).

Original entry on oeis.org

14, 36, 736, 8736, 48736, 948736, 2948736, 32948736, 432948736, 3432948736, 53432948736, 353432948736, 5353432948736, 75353432948736, 75353432948736, 5075353432948736, 15075353432948736, 615075353432948736, 8615075353432948736, 98615075353432948736, 98615075353432948736

Offset: 1

Jon E. Schoenfield, Apr 23 2007

			2^14 = 16384 and 14 end with the same single digit 4, thus a(1) = 14.

Jon E. Schoenfield, Table of n, a(n) for n = 1..81

See A121319, the main entry for this sequence, for further information.
Same as A109405 except for the initial term (14). - Max Alekseyev, May 11 2007
Cf. A007185, A016090, A003226, A035383, A064540, A064541, A109405, A121319, A206636.

A121319 a(n) is the smallest number k such that k and 2^k have the same last n digits. Here k must have at least n digits (cf. A113627).

Original entry on oeis.org

14, 36, 736, 8736, 48736, 948736, 2948736, 32948736, 432948736, 3432948736, 53432948736, 353432948736, 5353432948736, 75353432948736, 1075353432948736, 5075353432948736, 15075353432948736, 615075353432948736, 8615075353432948736, 98615075353432948736

Offset: 1

Tanya Khovanova, Aug 25 2006

			2^14 = 16384 and 14 end with the same single digit 4, thus a(1) = 14.

Jon E. Schoenfield, Table of n, a(n) for n = 1..80
Jon E. Schoenfield, Excel program

Cf. A007185, A016090, A003226, A035383, A064540, A064541, A109405.

f[n_] := Block[{k = If[n == 1, 2, 10], m = 10^n}, While[ PowerMod[2, k, m] != Mod[k, m], k += 2]; k]; Do[ Print@f@n, {n, 9}] (* Robert G. Wilson v *)

A121319(n) = { local(k,tn); tn=10^n ; forstep(k=2,1000000000,2, if ( k % tn == (2^k) % tn, return(k) ; ) ; ) ; return(0) ; } { for(n = 1,13, print( A121319(n)) ; ) ; } \\ R. J. Mathar, Aug 27 2006

If A109405(n) has n digits, a(n) = A109405(n), otherwise a(n) = A109405(n) + 10^n. - Max Alekseyev, May 05 2007

a(6)-a(9) from Robert G. Wilson v and Jon E. Schoenfield, Aug 26 2006
a(10) from Robert G. Wilson v, Sep 26 2006
a(11)-a(16) from Alexander Adamchuk, Jan 28 2007
a(16) corrected by Max Alekseyev, Apr 12 2007

A206636 a(n) = 2^^(n+2) modulo 10^n, where ^^ denotes a power tower (see A133612).

Original entry on oeis.org

6, 36, 736, 8736, 48736, 948736, 2948736, 32948736, 432948736, 3432948736, 53432948736, 353432948736, 5353432948736, 75353432948736, 75353432948736, 5075353432948736, 15075353432948736, 615075353432948736, 8615075353432948736, 98615075353432948736, 98615075353432948736, 8098615075353432948736

Offset: 1

Marco Ripà, Feb 10 2012

nonn

Backward concatenation of A133612.

For all m>n+1, 2^^m == 2^^(n+2) (mod 10^n). Hence, each term represents the trailing decimal digits of 2^^m for every sufficiently large m.

M. Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
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" from Vardi

Cf. A133612, A113627, A109405, A133613, A133614, A133615, A133616, A133617, A133618, A133619, A183613, A144539, A144540, A144541, A144542, A144543, A144544.

(* first load all lines of Super Power Mod by Ilan Vardi from the hyper-link, then *) $RecursionLimit = 2^14;  a[n_] := SuperPowerMod[2, n +2, 10^n]; Array[a, 22] (* Robert G. Wilson v, Apr 20 2020 *)

a(n) = A014221(n+3) mod (10^n).

For n>1, a(n) = 2^a(n-1) mod 10^n.

cp's OEIS Frontend

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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A064541 Numbers k such that 2^k ends in k.

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A113627 a(n) is the smallest number k such that k and 2^k have the same last n digits. Here k may have fewer than n digits and can be padded with leading zeros (cf. A121319).

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A121319 a(n) is the smallest number k such that k and 2^k have the same last n digits. Here k must have at least n digits (cf. A113627).

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A206636 a(n) = 2^^(n+2) modulo 10^n, where ^^ denotes a power tower (see A133612).

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