A306686 -id:A306686 - cp's OEIS frontend

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).

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

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

A290788 Values of n such that 6^n ends in n, or expomorphic numbers in "base" 6.

Original entry on oeis.org

6, 56, 656, 8656, 38656, 238656, 7238656, 47238656, 447238656, 7447238656, 27447238656, 227447238656, 3227447238656

Offset: 1

Bernard Schott, Aug 10 2017

Definition: For positive integers b (as base) and n, the positive integer (allowing initial 0's) a(n) is expomorphic relative to base b (here 6) if a(n) has exactly n decimal digits and if b^a(n) == a(n) (mod 10^n) or, equivalently, b^a(n) ends in a(n). [See Crux Mathematicorum link.]

			6^6 = 46656 ends in 6, so 6 is a term.
6^56 = ...656 ends in 56, so 56 is another term.

Charles W. Trigg, Problem 559, Crux Mathematicorum, page 192, Vol. 7, Jun. 81.

Cf. A064541 (base 2), A183613 (base 3), A288845 (base 4), A289138, A306570 (base 5), A306686 (base 9).

Cf. A003226 (automorphic numbers), A033819 (trimorphic numbers).

Select[Range[10^6], PowerMod[6, #, 10^(1 + Floor@ Log10[#])] == # &] (* Michael De Vlieger, Apr 13 2021 *)

is(n)=my(m=10^#digits(n)); Mod(6,m)^n==n \\ Charles R Greathouse IV, Aug 10 2017

a(6)-a(9) from Charles R Greathouse IV, Aug 10 2017

a(10)-a(13) from Chai Wah Wu, Apr 13 2021

A324017 Square array A(m,n) (m>=1, n>=1) read by antidiagonals: A(m,n) = (2n - 1)^^m mod (2n)^m (see Comments for definition of ^^).

Original entry on oeis.org

1, 3, 1, 5, 11, 1, 7, 29, 59, 1, 9, 55, 29, 59, 1, 11, 89, 119, 1109, 827, 1, 13, 131, 289, 3703, 3701, 2875, 1, 15, 181, 563, 5289, 7799, 34805, 15163, 1, 17, 239, 965, 16115, 45289, 138871, 128117, 31547, 1, 19, 305, 1519, 25661, 57587, 745289, 1711735, 687989, 97083, 1

Offset: 1

Davis Smith, Mar 28 2019

Tetration (x^^n) is defined as x^^0 = 1 and x^^n = x^(x^^(n - 1)). Another way to put this is that x^^n = x^x^x^...x with n x's.
Conjecture: For any three integers (greater than 1), m, n, and k, such that (2*n - 1)^^m == k (mod (2*n)^m), (2*n - 1)^k == k (mod (2*n)^m). For example, 5^^2 == 29 (mod 6^2) and 5^29 == 29 (mod 6^2).
Conjecture: For n > 1 and m >= 2, floor(((2*n - 1)^^m)/(2*n)) ==  2*(n - 1) (mod 2*n). For example, floor((13^^3)/14) == 12 (mod 14) and floor((15^^4)/16) == 14 (mod 16).
Conjecture: For m > 1, where (2*n - 1)^^m == j (mod (2*n)^(m + 1)), A(m + 1,n) = j. For example, A(6,3) = 563 and A(6,4) = 16115; 11^^3 == 563 (mod 12^3) and 11^^3 == 16115 (mod 12^4).

			Square array A(m,n) begins:
  \n  1     2      3       4        5          6         7          8 ...
  m\
   1| 1     3      5       7        9         11        13         15 ...
   2| 1    11     29      55       89        131       181        239 ...
   3| 1    59     29     119      289        563       965       1519 ...
   4| 1    59   1109    3703     5289      16115     25661      13807 ...
   5| 1   827   3701    7799    45289      57587    332989     669167 ...
   6| 1  2875  34805  138871   745289    1799411   4635581     669167 ...
   7| 1 15163 128117 1711735  2745289   25687283  49812797   67778031 ...
   8| 1 31547 687989 8003191 92745289  419837171 155226301 3557438959 ...
.
Examples of columns in this array:
A(m,1) = A000012(m - 1).
A(m,5) = A306686(m) with a note about how this sequence repeats terms rather than skipping.
Examples of rows in this array:
A(1,n) = A005408(n - 1).
A(2,n) = A082108(n - 1).

Charles W. Trigg, Problem 559, Crux Mathematicorum, page 192, Vol. 7, Jun. 81.
Eric Weisstein's World of Mathematics,Power Tower.
Wikipedia, Tetration.

Cf. A000012, A005408, A082108, A306686.

tetrmod(b,n,m)=my(t=b);i=0;while(i<=n, i++&&if(i>1, t=lift(Mod(b,m)^t), t)); t
tetrmatrix(lim)= matrix(lim,lim,x,y,tetrmod((2*y)-1,x,(2*y)^x))

A351410 Numbers m such that the decimal representation of 8^m ends in m.

Original entry on oeis.org

56, 856, 5856, 25856, 225856, 5225856, 95225856, 895225856, 6895225856, 16895225856, 416895225856, 5416895225856, 35416895225856, 7035416895225856, 77035416895225856, 577035416895225856, 1577035416895225856, 21577035416895225856, 521577035416895225856, 1521577035416895225856, 81521577035416895225856

Offset: 1

Bernard Schott, Feb 10 2022

The Crux Mathematicorum link calls these numbers "expomorphic" relative to "base" b, with here b = 8.
Under that definition, the term after a(13) = 35416895225856 is not "035416895225856" or "35416895225856" but a(14) = 7035416895225856.
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).
This conjecture is true. See A133618. - David A. Corneth, Feb 10 2022

			8^56 = 374144419156711147060143317175368453031918731001856, so 56 is a term.
8^856 = ...5856 ends in 856, so 856 is another term.

Charles W. Trigg, Problem 559, Crux Mathematicorum, pp. 192-194, Vol. 7, Jun. 1981.

Cf. A064541, A183613, A288845, A306570, A290788, A321970, this sequence, A306686, A289138.
Cf. A003226 (automorphic numbers), A033819 (trimorphic numbers).
Cf. A133618 (leading digits).

a(7)-a(8) from Michel Marcus, Feb 10 2022

More terms from David A. Corneth, Feb 10 2022

cp's OEIS Frontend

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

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PARI

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A290788 Values of n such that 6^n ends in n, or expomorphic numbers in "base" 6.

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Mathematica

PARI

Extensions

A324017 Square array A(m,n) (m>=1, n>=1) read by antidiagonals: A(m,n) = (2*n - 1)^^m mod (2*n)^m (see Comments for definition of ^^).

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Keywords

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Examples

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Programs

PARI

A351410 Numbers m such that the decimal representation of 8^m ends in m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A324017 Square array A(m,n) (m>=1, n>=1) read by antidiagonals: A(m,n) = (2n - 1)^^m mod (2n)^m (see Comments for definition of ^^).