A133615 -id:A133615 - cp's OEIS frontend

A144542 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 14^A(k) == A(k) mod 10^k.

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

Offset: 0

N. J. A. Sloane, Dec 20 2008

			633205765773282307762880481730639884053299234674143456126141831703991362480509...

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, A133618, A133619, A144539, A144540, A144541, A144543, A144544.

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

a(68) onward from Robert G. Wilson v, Mar 06 2014

A144543 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 15^A(k) == A(k) mod 10^k.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 20 2008

			573958083567096086449346119283793862477858654472393049431441904930012219852452...

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, A133618, A133619, A144539, A144540, A144541, A144542, A144544.

(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[15, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)

a(68) onward from Robert G. Wilson v, Mar 06 2014

A144544 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 16^A(k) == A(k) mod 10^k.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 20 2008

			616514092059405701876632862258462088380056998252117853367321783700266620705906...

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, A133618, A133619, A144539, A144540, A144541, A144542, A144543.

(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[16, n + 1, 10^n]; Reverse@ IntegerDigits@ f@ 105 (* Robert G. Wilson v, Mar 06 2014 *)

a(68) onward from Robert G. Wilson v, Mar 06 2014

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.

A306570 Values of n such that 5^n ends in n, or expomorphic numbers relative to "base" 5.

Original entry on oeis.org

5, 25, 125, 3125, 203125, 8203125, 408203125, 8408203125, 18408203125, 618408203125, 2618408203125, 52618408203125, 152618408203125, 3152618408203125, 93152618408203125, 493152618408203125, 7493152618408203125, 17493152618408203125, 117493152618408203125, 7117493152618408203125, 87117493152618408203125

Offset: 1

Bernard Schott, Feb 24 2019

Definition: For positive integers b (as base) and n, the positive integer (allowing initial zeros) k(n) is expomorphic relative to base b (here 5) if k(n) has exactly n decimal digits and if b^k(n) == k(n) (mod 10^n) or, equivalently, b^k(n) ends in k(n). [See Crux Mathematicorum link.]
For sequences in the OEIS, no term is allowed to begin with a digit 0 (except for the 1-digit number 0 itself). However, in the problem as defined in the Crux Mathematicorum article, leading 0 digits are allowed; under that definition a(n) = k(n) until the first k(n) which begins with digit 0. When k(n) begins with 0, then, a(n) is the next term of the sequence k(n) which doesn't begin with digit 0.
Under that definition, the term after a(4) = 3125 is not "03125" but a(5) = 203125. [Comments from Jon E. Schoenfield in A288845 and discussion with Rémy Sigrist]
Conjecture: if k(n) is expomorphic relative to "base" b, then, the next one in the sequence, k(n+1), consists of the last n+1 digits of b^k(n).
a(n) is the backward concatenation of A133615(0) through A133615(n-1). So, a(1) is 5, a(2) is 25, and so on, with recognition of the comments about the OEIS and terms beginning with 0 (for example, when n = 5, A133615(n-1) = 0, so the next nonzero digit is concatenated as well, reducing the amount subtracted from n by 1). - Davis Smith Mar 07 2019

			5^5 = 25 ends in 5, so 5 is a term; 5^25 = ...125 ends in 25, so 25 is another term.

Davis Smith, Table of n, a(n) for n = 1..894
Charles W. Trigg, Problem 559, Crux Mathematicorum, page 192, Vol. 7, Jun. 81.

Cf. A064541 (base 2), A183613 (base 3), A288845 (base 4), A290788 (base 6), A321970 (base 7), A306686 (base 9), A289138 (smallest expomorphic number in base n).
Cf. A003226 (automorphic numbers), A033819 (trimorphic numbers).
Cf. A133615 (leading digits).

is(n) = my(t=#digits(n)); lift(Mod(5, 10^t)^n)==n
for(n=1, oo, my(x=n*5); if(lift(Mod(5, 10)^x)==x%10, if(is(x), print1(x, ", ")))) \\ Felix Fröhlich, Feb 24 2019

tetrmod(b,n,m)=my(t=b); for(i=1, n, if(i>1, t=lift(Mod(b,m)^t), t)); t
for(n=1, 21,if(tetrmod(5,n,10^n)!=tetrmod(5,n-1,10^(n-1)),print1(tetrmod(5,n,10^(n-1)),", "))) \\ Davis Smith, Mar 09 2019

a(5)-a(7) from Felix Fröhlich, Feb 24 2019
a(8) from Michel Marcus, Mar 02 2019
a(9)-a(21) from Davis Smith, Mar 07 2019

A318478 Decimal digits such that for all k>=1, the number A(k) := Sum_{n = 0..k-1 } a(n)*10^n satisfies the congruence 1984^A(k) == A(k) (mod 10^k).

Original entry on oeis.org

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

Offset: 1

Marco Ripà, Aug 26 2018

10-adic expansion of the iterated exponential 1984^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n>=9, 1984^^n(mod 10^8) == 98703616.

1984^^n, for any n>=188, appears in M. Ripà's book "La strana coda della serie n^n^...^n", where the author took his birth year (1984), as a random base in order to prove some general properties about tetration, and calculating 1984^^n(mod 10^187) as a test for his paper-and-pencil procedure.

			1984^^1984 (mod 10^8) == 98703616.
Thus, 1984^^1984 = ...61630789307145912032948400109045102(...)7490335.
Consider the sequence 1984^^n: 1984, 1984^1984, 1984^(1984^1984), ... From 1984^^3 onwards, all terms end with the digits 16. 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. 78-79. ISBN 978-88-6178-789-6.
Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.

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
Wikipedia, Graham's number
Wikipedia, Tetration

Cf. A133612, A133613, A133614, A133615, A133616, A133617, A133618, A133619, A144539, A144540, A144541, A144542, A144543, A144544, A317824, A317903, A317905.

cp's OEIS Frontend

A144542 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 14^A(k) == A(k) mod 10^k.

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A144543 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 15^A(k) == A(k) mod 10^k.

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A144544 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 16^A(k) == A(k) mod 10^k.

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

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A306570 Values of n such that 5^n ends in n, or expomorphic numbers relative to "base" 5.

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A318478 Decimal digits such that for all k>=1, the number A(k) := Sum_{n = 0..k-1 } a(n)*10^n satisfies the congruence 1984^A(k) == A(k) (mod 10^k).

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