A003226 -id:A003226 - cp's OEIS frontend

A274463 Subsequence of the automorphic numbers (A003226) with initial term 5 and such that a(n+1) ends with the digits of a(n).

5, 25, 625, 12890625, 6259918212890625, 4106619977392256259918212890625, 80863811000557423423230896109004106619977392256259918212890625

Offset: 1

Colin Barker, Jun 24 2016

An automorphic number is a number whose square ends in the same digits as the number itself.

The next term has 126 digits.

Colin Barker, Table of n, a(n) for n = 1..10
Wikipedia, Automorphic number

Cf. A003226, A274464.

seq(m, maxn) = L=List(); for(j=1, maxn, listput(L, m); m=(3*m^2-2*m^3)%10 ^ (2*sizedigit(m))); Vec(L)
seq(5, 8)

A274464 Subsequence of the automorphic numbers (A003226) with initial term 6 and such that a(n+1) ends with the digits of a(n).

Original entry on oeis.org

6, 76, 139376, 114087109376, 792415373740081787109376, 88398678125844615295893380022607743740081787109376, 3724919229963099270422168663257939520419136188999442576576769103890995893380022607743740081787109376

Offset: 1

Colin Barker, Jun 24 2016

An automorphic number is a number whose square ends in the same digits as the number itself.

The next term has 200 digits.

Colin Barker, Table of n, a(n) for n = 1..10
Wikipedia, Automorphic number

Cf. A003226, A274463.

seq(m, maxn) = L=List(); for(j=1, maxn, listput(L, m); m=(3*m^2-2*m^3)%10 ^ (2*sizedigit(m))); Vec(L)
seq(6, 8)

A307104 a(n) is the number which, when concatenated with A003226(n), the n-th automorphic number, gives (A003226(n))^2.

Original entry on oeis.org

0, 0, 2, 3, 6, 57, 141, 390, 8790, 82128, 11963, 793212, 835571, 5054322, 1661682, 75880433, 45322418, 619541169, 319375992, 6745157241, 3317093849, 66891312600, 843114912509, 9837094694375, 16065496578813, 35901922360062, 67557477392256, 547721051611007

Offset: 1

Christopher Hohl, Mar 24 2019

Let na and nb represent the indices of the preceding and next A003226(n)'s beginning with a 9, and where (na - nb) >= 3 (note that the first such 'zone' begins with an exception for which the index A003226(na) = 1). Then for na < n < nb and such that n == (na + 1) mod 2, it appears that A003226(n) - a(n) = A003226(n+1) - a(n+1) = k.

In such cases, it also appears that a(n)*a(n+1) = k^2 - k.

			For n=4, A003226(4)=6, (A003226(4))^2=36. So a(4)=3.
For n=13, A003226(13)=2890625, (A003226(13))^2=8355712890625. So a(13)=835571.

Cf. A003226, A055642.

auto(n) = {n<3 & return(n-1); my(i=10, j=10, b=5, c=6, a=b); for( k=4, n, while(b<=a, b=b^2%i*=10); while(c<=a, c=(2-c)*c%j*=10); a=min(b, c)); a; } \\ A003226
a(n) = {my(m = auto(n), dm = digits(m), dm2 = digits(m^2)); fromdigits(vector(#dm2 - #dm, k, dm2[k]));} \\ Michel Marcus, May 18 2019

a(n) = A003226(n)*(A003226(n) - 1) / 10^A055642(A003226(n)).

A018247 The 10-adic integer x = ...8212890625 satisfying x^2 = x.

Original entry on oeis.org

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

Offset: 0

Yoshihide Tamori (yo(AT)salk.edu)

The 10-adic numbers a and b defined in this sequence and A018248 satisfy a^2=a, b^2=b, a+b=1, ab=0. - Michael Somos

			x = ...0863811000557423423230896109004106619977392256259918212890625.

W. W. R. Ball, Mathematical Recreations & Essays, N.Y. Macmillan Co, 1947.
V. deGuerre and R. A. Fairbairn, Jnl. Rec. Math., No. 3, (1968), 173-179.
M. Kraitchik, Sphinx, 1935, p. 1.

Eric M. Schmidt, Table of n, a(n) for n = 0..9999
Anthony Edey, Automorphic numbers, taken from Madachy's Mathematical Recreations, Dover 1979.
V. deGuerre and R. A. Fairbairn, Automorphic numbers, Jnl. Rec. Math., 1 (No. 3, 1968), 173-179.
MathOverflow, Distribution of digits of pq-adic idempotents (aka “automorphic numbers”), 2014.
Eric Weisstein's World of Mathematics, Automorphic numbers.
Index entries for sequences related to automorphic numbers

A007185 gives associated automorphic numbers.
Cf. A018248, A033819, A003226.
The difference between A018248 & this sequence is A075693 and their product is A075693.
The six examples given by deGuerre and Fairbairn are A055620, A054869, A018247, A018248, A259468, A259469.

a = {5}; f[n_] := Block[{k = 0, c}, While[c = FromDigits[Prepend[a, k]]; Mod[c^2, 10^n] != c, k++ ]; a = Prepend[a, k]]; Do[ f[n], {n, 2, 105}]; Reverse[a]
With[{n = 150}, Reverse[IntegerDigits[PowerMod[5, 2^n, 10^n]]]] (* IWABUCHI Yu(u)ki, Feb 16 2024 *)

a(n)=local(t=5);for(k=1,n+1,t=t^2%10^k);t\10^n \\ Paul D. Hanna, Jul 08 2006

Vecrev(digits(lift(chinese(Mod(1, 2^100), Mod(0, 5^100))))) \\ Seiichi Manyama, Aug 07 2019

x = 10-adic lim_{n->oo} 5^(2^n) mod 10^(n+1). - Paul D. Hanna, Jul 08 2006

More terms from David W. Wilson

Edited by David W. Wilson, Sep 26 2002

A008851 Congruent to 0 or 1 mod 5.

Original entry on oeis.org

0, 1, 5, 6, 10, 11, 15, 16, 20, 21, 25, 26, 30, 31, 35, 36, 40, 41, 45, 46, 50, 51, 55, 56, 60, 61, 65, 66, 70, 71, 75, 76, 80, 81, 85, 86, 90, 91, 95, 96, 100, 101, 105, 106, 110, 111, 115, 116, 120, 121, 125, 126, 130, 131, 135, 136, 140, 141, 145, 146, 150, 151

Offset: 1

N. J. A. Sloane

Numbers k that have the same last digit as k^2.

L. E. Dickson, History of the Theory of Numbers, I, p. 459.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001622, A003226, A045953, A046831, A046851, A086457.

a008851 n = a008851_list !! (n-1)
a008851_list = [10*n + m | n <- [0..], m <- [0,1,5,6]]
-- Reinhard Zumkeller, Jul 27 2011

[n: n in [0..200] | n mod 5 in {0, 1}]; // Vincenzo Librandi, Nov 17 2014

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-2]+5 od: seq(a[n], n=0..61); # Zerinvary Lajos, Mar 16 2008

Select[Range[0, 151], MemberQ[{0, 1}, Mod[#, 5]] &] (* T. D. Noe, Mar 31 2013 *)

a(n) = 5*(n\2)+bitand(n,1); /* Joerg Arndt, Mar 31 2013 */

a(n) = floor((5/3)*floor(3*(n-1)/2)); /* Joerg Arndt, Mar 31 2013 */

a(n) = 5*n - a(n-1) - 9, n >= 2. - Vincenzo Librandi, Nov 18 2010 [Corrected for offset by David Lovler, Oct 10 2022]
G.f.: x^2*(1+4*x) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 07 2011
a(n+1) = Sum_{k>=0} A030308(n,k)*A146523(k). - Philippe Deléham, Oct 17 2011
a(n) = floor((5/3)*floor(3*(n-1)/2)). - Clark Kimberling, Jul 04 2012
a(n) = (10*n - 13 - 3*(-1)^n)/4. - Robert Israel, Nov 17 2014 [Corrected by David Lovler, Sep 21 2022]
E.g.f.: 4 + ((10*x - 13)*exp(x) - 3*exp(-x))/4. - David Lovler, Sep 11 2022
Sum_{n>=2} (-1)^n/a(n) = sqrt(1+2/sqrt(5))*Pi/10 + log(phi)/(2*sqrt(5)) + log(5)/4, where phi is the golden ratio (A001622). - Amiram Eldar, Oct 12 2022

Offset corrected by Reinhard Zumkeller, Jul 27 2011

A007185 Automorphic numbers ending in digit 5: a(n) = 5^(2^n) mod 10^n.

Original entry on oeis.org

5, 25, 625, 625, 90625, 890625, 2890625, 12890625, 212890625, 8212890625, 18212890625, 918212890625, 9918212890625, 59918212890625, 259918212890625, 6259918212890625, 56259918212890625, 256259918212890625, 2256259918212890625, 92256259918212890625

Offset: 1

N. J. A. Sloane

Conjecture: For any m coprime to 10 and for any k, the density of n such that a(n) == k (mod m) is 1/m. - Eric M. Schmidt, Aug 01 2012

a(n) is the unique positive integer less than 10^n such that a(n) is divisible by 5^n and a(n) - 1 is divisible by 2^n. - Eric M. Schmidt, Aug 18 2012

			625 is in the sequence because 625^2 = 390625, which ends in 625.
90625 is in the sequence because 90625^2 = 8212890625, which ends in 90625.
90635 is not in the sequence because 90635^2 = 8214703225, which does not end in 90635.

V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.
R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2 (No. 3, 1969), 170-174.
Jan Gullberg, Mathematics, From the Birth of Numbers, W. W. Norton & Co., NY, page 253-4.
Ya. I. Perelman, Algebra can be fun, pp. 97-98.
C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
C. P. Schut, Idempotents, Report AM-R9101, Centre for Mathematics and Computer Science, Amsterdam, 1991. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Automorphic Number
Xiaolong Ron Yu, Curious Numbers, Pi Mu Epsilon Journal, Spring 1999, pp. 819-823.
Index entries for sequences related to automorphic numbers

A018247 gives the associated 10-adic number.

A003226 = {0, 1} union (this sequence) union A016090.

[Modexp(5, 2^n, 10^n): n in [1..30]]; // Vincenzo Librandi, Jun 11 2016

a:= n-> 5&^(2^n) mod 10^n: seq(a(n), n=1..25);  # Alois P. Heinz, Mar 11 2018

Table[PowerMod[5, 2^n, 10^n], {n, 25}] (* Vincenzo Librandi, Jun 11 2016 *)

A007185(n)=lift(Mod(5,10^n)^2^n)  \\ M. F. Hasler, Dec 05 2012

[crt(1, 0, 2^n, 5^n) for n in range(1, 1001)] # Eric M. Schmidt, Aug 18 2012

a(n) = 5^(2^n) mod 10^n.
a(n)^2 == a(n) (mod 10^n), that is, a(n) is an idempotent in Z[10^n].
a(n+1) = a(n)^2 mod 10^(n+1). - Eric M. Schmidt, Jul 28 2012
a(2n) = (3*a(n)^2 - 2*a(n)^3) mod 10^(2n). - Sylvie Gaudel, Mar 10 2018

More terms from David W. Wilson
Edited by David W. Wilson, Sep 26 2002
Further edited by N. J. A. Sloane, Jul 21 2010
Comment moved to name by Alonso del Arte, Mar 10 2018

A016090 a(n) = 16^(5^n) mod 10^n: Automorphic numbers ending in digit 6, with repetitions.

Original entry on oeis.org

6, 76, 376, 9376, 9376, 109376, 7109376, 87109376, 787109376, 1787109376, 81787109376, 81787109376, 81787109376, 40081787109376, 740081787109376, 3740081787109376, 43740081787109376, 743740081787109376, 7743740081787109376, 7743740081787109376

Offset: 1

Robert G. Wilson v, David W. Wilson

Also called congruent numbers.
a(n)^2 == a(n) (mod 10^n), that is, a(n) is idempotent of Z[10^n].
Conjecture: For any m coprime to 10 and for any k, the density of n such that a(n) == k (mod m) is 1/m. - Eric M. Schmidt, Aug 01 2012
a(n) is the unique positive integer less than 10^n such that a(n) is divisible by 2^n and a(n) - 1 is divisible by 5^n. - Eric M. Schmidt, Aug 18 2012

			a(5) = 09376 because 09376^2 == 87909376 ends in 09376.

R. Cuculière, Jeux Mathématiques, in Pour la Science, No. 6 (1986), 10-15.
V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.
R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2 (No. 3, 1969), 170-174.
Jan Gullberg, Mathematics, From the Birth of Numbers, W. W. Norton & Co., NY, page 253-4.
Ya. I. Perelman, Algebra can be fun, pp. 97-98.
A. M. Robert, A Course in p-adic Analysis, Springer, 2000; see pp. 63, 419.
C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
C. P. Schut, Idempotents, Report AM-R9101, Centre for Mathematics and Computer Science, Amsterdam, 1991. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Automorphic Number
Xiaolong Ron Yu, Curious Numbers, Pi Mu Epsilon Journal, Spring 1999, pp. 819-823.
Index entries for sequences related to automorphic numbers

A018248 gives the associated 10-adic number.

A003226 = {0, 1} union A007185 union (this sequence).

List([1..22], n->PowerModInt(16,5^n,10^n)); # Muniru A Asiru, Mar 20 2018

[Modexp(16, 5^n, 10^n): n in [1..30]]; // Bruno Berselli, Mar 13 2018

[seq(16 &^ 5^n mod 10^n, n=1..22)]; # Muniru A Asiru, Mar 20 2018

Array[PowerMod[16, 5^#, 10^#] &, 18] (* Michael De Vlieger, Mar 13 2018 *)

A016090(n)=lift(Mod(6,10^n)^5^(n-1)) \\ M. F. Hasler, Dec 05 2012, edited Jan 26 2020

[crt(0, 1, 2^n, 5^n) for n in range(1, 1001)] # Eric M. Schmidt, Aug 18 2012

a(n) = 16^(5^n) mod 10^n.
a(n+1) == 2*a(n) - a(n)^2 (mod 10^(n+1)). - Eric M. Schmidt, Jul 28 2012
a(n) = 6^(5^n) mod 10^n. - Sylvie Gaudel, Feb 17 2018
a(2*n) = (3*a(n)^2 - 2*a(n)^3) mod 10^(2*n). - Sylvie Gaudel, Mar 12 2018
a(n) = 6^5^(n-1) mod 10^n. - M. F. Hasler, Jan 26 2020
a(n) = 2^(10^n) mod 10^n for n >= 2. - Peter Bala, Nov 10 2022

Edited by David W. Wilson, Sep 26 2002

Definition corrected by M. F. Hasler, Dec 05 2012

A018248 The 10-adic integer x = ...1787109376 satisfies x^2 = x.

Original entry on oeis.org

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

Offset: 0

Yoshihide Tamori (yo(AT)salk.edu)

The 10-adic numbers a and b defined in A018247 and this sequence satisfy a^2=a, b^2=b, a+b=1, ab=0. - Michael Somos

			x equals the limit of the (n+1) trailing digits of 6^(5^n):
6^(5^0)=(6), 6^(5^1)=77(76), 6^(5^2)=28430288029929701(376), ...
x = ...9442576576769103890995893380022607743740081787109376.
From _Peter Bala_, Nov 05 2022: (Start)
Trailing digits of 2^(10^n), 4^(10^n) and 6^(10^n) for n = 5:
2^(10^5) = ...9883(109376);
4^(10^5) = ...7979(109376);
6^(10^5) = ...4155(109376). (End)

W. W. R. Ball, Mathematical Recreations & Essays, N.Y. Macmillan Co, 1947.
R. Cuculière, Jeux Mathématiques, in Pour la Science, No. 6 (1986), 10-15.
V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.
M. Kraitchik, Sphinx, 1935, p. 1.
A. M. Robert, A Course in p-adic Analysis, Springer, 2000; see pp. 63, 419.

Seiichi Manyama, Table of n, a(n) for n = 0..9999 (terms 0..999 from Paul D. Hanna).
Anonymous, Automorphic numbers (2)
Peter Bala, A note on A018248
V. deGuerre and R. A. Fairbairn, Automorphic numbers, Jnl. Rec. Math., 1 (No. 3, 1968), 173-179.
MathOverflow, Distribution of digits of pq-adic idempotents (a.k.a. "automorphic numbers"), 2014.
Eric Weisstein's World of Mathematics, Automorphic numbers (1)
Index entries for sequences related to automorphic numbers

A016090 gives associated automorphic numbers.
Cf. A018247, A033819.
The difference between this sequence & A018247 is A075693 and their product is A075693.
Cf. A120817, A120818, A091664.
The six examples given by deGuerre and Fairbairn are A055620, A054869, A018247, A018248, A259468, A259469.

a := proc (n) option remember; if n = 1 then 2 else irem(a(n-1)^10, 10^n) end if; end proc:
# display the digits of a(100) from right to left
S := convert(a(100), string):
with(ListTools):
the_List := [seq(parse(S[i]), i = 1..length(S))]:
Reverse(the_List); # Peter Bala, Nov 04 2022

b = {6}; g[n_] := Block[{k = 0, c}, While[c = FromDigits[Prepend[b, k]]; Mod[c^2, 10^n] != c, k++ ]; b = Prepend[b, k]]; Do[ g[n], {n, 2, 105}]; Reverse[b]
With[{n = 150}, Reverse[IntegerDigits[PowerMod[16, 5^n, 10^n]]]] (* IWABUCHI Yu(u)ki, Feb 16 2024 *)

{a(n)=local(b=6,v=[]);for(k=1,n+1,b=b^5%10^k;v=concat(v,(10*b\10^k)));v[n+1]} \\ Paul D. Hanna, Jul 06 2006

Vecrev(digits(lift(chinese(Mod(0, 2^100), Mod(1, 5^100))))) \\ Seiichi Manyama, Aug 07 2019

x = r^4 where r=...1441224165530407839804103263499879186432 (A120817). x = 10-adic limit_{n->oo} 6^(5^n). - Paul D. Hanna, Jul 06 2006

For n >= 2, the final n+1 digits of either 2^(10^n), 4^(10^n) or 6^(10^n), when read from right to left, give the first n+1 entries in the sequence. - Peter Bala, Nov 05 2022

More terms from David W. Wilson

Edited by David W. Wilson, Sep 26 2002

A018834 Numbers k such that the decimal expansion of k^2 contains k as a substring.

Original entry on oeis.org

0, 1, 5, 6, 10, 25, 50, 60, 76, 100, 250, 376, 500, 600, 625, 760, 1000, 2500, 3760, 3792, 5000, 6000, 6250, 7600, 9376, 10000, 14651, 25000, 37600, 50000, 60000, 62500, 76000, 90625, 93760, 100000, 109376, 250000, 376000, 495475, 500000, 505025

Offset: 1

David W. Wilson

			25^2 = 625 which contains 25.
3792^2 = 14_3792_64, 14651^2 = 2_14651_801.

Giovanni Resta, Table of n, a(n) for n = 1..200 (first 126 terms from David W. Wilson)

Cf. A003226, A046831, A046851, A045953.
Cf. A000290. Supersequence of A029943.
Cf. A018826 (base 2), A018827 (base 3), A018828 (base 4), A018829 (base 5), A018830 (base 6), A018831 (base 7), A018832 (base 8), A018833 (base 9).
Cf. A029942 (cubes), A075904 (4th powers), A075905 (5th powers).

import Data.List (isInfixOf)
a018834 n = a018834_list !! (n-1)
a018834_list = filter (\x -> show x `isInfixOf` show (x^2)) [0..]
-- Reinhard Zumkeller, Jul 27 2011

Select[Range[510000], MemberQ[FromDigits /@ Partition[IntegerDigits[#^2], IntegerLength[#], 1], #] &] (* Jayanta Basu, Jun 29 2013 *)
Select[Range[0,510000],StringPosition[ToString[#^2],ToString[#]]!={}&] (* Ivan N. Ianakiev, Oct 02 2016 *)

from itertools import count, islice
def A018834_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:str(n) in str(n**2), count(max(startvalue,0)))
A018834_list = list(islice(A018834_gen(),20)) # Chai Wah Wu, Apr 04 2023

A237583 Automorphic numbers: n^2 ends with n in base 6.

Original entry on oeis.org

0, 1, 3, 4, 9, 28, 81, 136, 1216, 6561, 16768, 29889, 76545, 203392, 636417, 1043200, 3995649, 6082048, 24151041, 36315136, 326481921, 689278977, 1487503360, 11573190657, 76876660737, 155240824833, 314944159744, 785129144320, 2035980763137, 4857090670593

Offset: 1

Eric M. Schmidt, Feb 09 2014

			From A201821:
a(3) = (3)_6 = 3 since 3^2 = 9 = (13)_6 ends with 3 in base 6.
a(4) = (4)_6 = 4 since 4^2 = 16 = (24)_6 ends with 4 in base 6.
a(5) = (13)_6 = 9 since 9^2 = 81 = (213)_6 ends with 13 in base 6.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A201821 (written in base 6), A003226, A201918, A201919, A201921, A201948.

isok(n) = ((n^2-n) % 6^(#digits(n, 6))) == 0; \\ Michel Marcus, Mar 08 2014

Sage
```
# See A003226.
```

cp's OEIS Frontend

A274463 Subsequence of the automorphic numbers (A003226) with initial term 5 and such that a(n+1) ends with the digits of a(n).

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A274464 Subsequence of the automorphic numbers (A003226) with initial term 6 and such that a(n+1) ends with the digits of a(n).

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A307104 a(n) is the number which, when concatenated with A003226(n), the n-th automorphic number, gives (A003226(n))^2.

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A018247 The 10-adic integer x = ...8212890625 satisfying x^2 = x.

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A008851 Congruent to 0 or 1 mod 5.

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A007185 Automorphic numbers ending in digit 5: a(n) = 5^(2^n) mod 10^n.

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A016090 a(n) = 16^(5^n) mod 10^n: Automorphic numbers ending in digit 6, with repetitions.

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