A018248 -id:A018248 - cp's OEIS frontend

A120817 10-adic integer x=...07839804103263499879186432 satisfying x^5 = x; also x^3 = -x = A120818; (x^2)^3 = x^2 = A091664; (x^4)^2 = x^4 = A018248.

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

Offset: 0

Paul D. Hanna, Jul 06 2006

			x equals the limit of the (n+1) trailing digits of 2^(5^n):
2^(5^0)=(2), 2^(5^1)=(32), 2^(5^2)=33554(432),
2^(5^3)=4253529586511730793292182592897102(6432), ...
x=...93839649523223304553032451441224165530407839804103263499879186432.
x^2=...0557423423230896109004106619977392256259918212890624 (A091664).
x^3=...6695446967548558775834469592160195896736500120813568 (A120818).
x^4=...9442576576769103890995893380022607743740081787109376 (A018248).
x^5=...3304553032451441224165530407839804103263499879186432 = x.

Seiichi Manyama, Table of n, a(n) for n = 0..9999 (terms 0..999 from Paul D. Hanna)
Patrick A. Thomas, Solutions to x^5=x up to base 100

x^5 = x: this sequence (...6432), A120818 (...3568), A290372 (...5807), A290373 (...2943), A290374 (...7057), A290375 (...4193).

Cf. A091664, A018248.

{a(n)=local(b=2,v=[]);for(k=1,n+1,b=b^5%10^k;v=concat(v,(10*b\10^k)));v[n+1]}

{a(n)=if(n<0, 0, lift(chinese(Mod(truncate( teichmuller(2+O(5^(n+1)))), 5^(n+1)), Mod(0, 2^(n+1))))\10^n)} /* Michael Somos, Oct 03 2006 */

x = 10-adic limit_{n->infinity} 2^(5^n).

A120818 10-adic integer x=...92160195896736500120813568 satisfying x^5 = x; also x^3 = -x = A120817; (x^2)^3 = x^2 = A091664; (x^4)^2 = x^4 = A018248.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Jul 06 2006

			x equals the limit of the (n+1) trailing digits of 8^(5^n):
8^(5^0)=(8), 8^(5^1)=327(68), 8^(5^2)=37778931862957161709(568), ...
x=...06160350476776695446967548558775834469592160195896736500120813568.
x^2=...0557423423230896109004106619977392256259918212890624 (A091664).
x^3=...3304553032451441224165530407839804103263499879186432 (A120817).
x^4=...9442576576769103890995893380022607743740081787109376 (A018248).
x^5=...6695446967548558775834469592160195896736500120813568 = x.

Seiichi Manyama, Table of n, a(n) for n = 0..9999 (terms 0..999 from Paul D. Hanna)

Cf. A120817, A091664, A018248.

{a(n)=local(b=8,v=[]);for(k=1,n+1,b=b^5%10^k;v=concat(v,(10*b\10^k)));v[n+1]}

x = 10-adic lim_{n->oo} 8^(5^n).

A075693 Difference between 10-adic numbers defined in A018248 & A018247.

Original entry on oeis.org

1, 5, -3, 9, -9, -7, 5, 7, 5, -7, 7, -9, -9, -1, 5, -3, -1, 5, 5, -9, 3, -5, -5, -9, -9, 7, -3, -3, 9, 7, 1, 9, 9, -9, 9, 7, -3, -9, -7, 9, 3, 5, 3, 5, 1, 3, 5, 1, -5, -1, -1, 9, 9, 9, 7, 7, -7, 3, -3, -7, 9, -7, -1, -9, 9, -1, -3, -3, 7, 5, -3, 9, 9, -9, -7, -9, 9, -1, -7, 3, -9, 5, 9, -7

Offset: 0

Robert G. Wilson v, Sep 26 2002

Numbers in A018247 and A018248 are known as automorphic numbers in base 10, meaning that the infinite integers a=(...256259918212890625) or b=(...743740081787109376) provides a nontrivial solution to x*x == x (mod any power of 10).
Read backwards so as to match their counterparts (A007185 & A016090), A018247(0)+A018248(0) = 11 & A018247(n)+A018248(n) = 9 for all n's > 0 and their product is A076308.
All entries must be odd.
Is the accumulative sum equally positive and negative, i.e. does the sum equal 0 infinitely often?

Seiichi Manyama, Table of n, a(n) for n = 0..9999

Cf. A018248 & A018247.

(* execute the programming in both A018247 & A018248 *) Reverse[b - a]
aa[n_] := For[t = 5; k = 1, True, k++, t = Mod[t^2, 10^k]; If[k == n, Return[ Quotient[t, 10^(n-1)]]]]; bb[n_] := Reap[ For[t = 6; k = 1, k <= n , k++, t = Mod[t^5, 10^k]; Sow[ Quotient[10*t, 10^k]]]][[2, 1, n]]; a[n_] := bb[n] - aa[n]; Table[a[n], {n, 1, 84}](* Jean-François Alcover, May 25 2012, after Paul D. Hanna *)

a(n) = A018248(n) - A018247(n). - Seiichi Manyama, Jul 26 2017

A076308 Product of 10-adic numbers defined in A018247 and A018248.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Oct 05 2002

Cf. A018247, A018248, A007185, A016090.

(* execute the programming in both A018247 & A018248 *) IntegerDigits[ FromDigits[ Reverse[a]]*FromDigits[ Reverse[b]]]

A003226 Automorphic numbers: m^2 ends with m.

Original entry on oeis.org

0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376, 59918212890625, 259918212890625, 740081787109376

Offset: 1

N. J. A. Sloane

Also called curious numbers.
For entries after the second, two successive terms sum up to a total having the form 10^n + 1. - Lekraj Beedassy, Apr 29 2005 [This comment is clearly wrong as stated. The sums of two consecutive terms are 1, 6, 11, 31, 101, 452, 1001, 10001, 100001, 200001, 1000001, 3781250, .... - T. D. Noe, Nov 14 2010]
If a d-digit number n is in the sequence, then so is 10^d+1-n. However, the same number can be 10^d+1-n for different n in the sequence (e.g., 10^3+1-376 = 10^4+1-9376 = 625), which spoils Beedassy's comment. - Robert Israel, Jun 19 2015
Substring of both its square and its cube not congruent to 0 (mod 10). See A029943. - Robert G. Wilson v, Jul 16 2005
a(n)^k ends with a(n) for k > 0; see also A029943. - Reinhard Zumkeller, Nov 26 2011
Apart from initial term, a subsequence of A046831. - M. F. Hasler, Dec 05 2012
This is also the sequence of numbers such that the n-th m-gonal number ends in n for any m == 0,4,8,16 (mod 20). - Robert Dawson, Jul 09 2018
Apart from 6, a subsequence of A301912. - Robert Dawson, Aug 01 2018

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 76, p. 26, Ellipses, Paris 2008.
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 253-254.
B. A. Naik, 'Automorphic numbers' in 'Science Today'(subsequently renamed '2001') May 1982 pp. 59, Times of India, Mumbai.
Ya. I. Perelman, Algebra can be fun, pp. 97-98.
Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Hoboken, 2005, p. 64.
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).

Chai Wah Wu, Table of n, a(n) for n = 1..1786 (terms n = 1..200 from Vincenzo Librandi)
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2.3 (1969), 170-174. (Annotated scanned copy)
V. deGuerre & R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1.3 (1968), 173-179
M. Beeler, R. W. Gosper, and R. Schroeppel, HAKMEM. MIT AI Memo 239, Feb 29 1972 (Item 59 provides a 40-digit member of this sequence).
Shyam Sunder Gupta, Elegance of Squares, Cubes, and Higher Powers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 2, 29-81.
W. Schneider, Automorphic Numbers
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
Wikipedia, Automorphic number
Xiaolong Ron Yu, Curious Numbers, Pi Mu Epsilon Journal, Spring 1999, pp. 819-823.
Index entries for sequences related to automorphic numbers

Cf. A008851, A018247, A018248, A018834, A033819, A035383, A046831, A052228.

import Data.List (isSuffixOf)
a003226 n = a003226_list !! (n-1)
a003226_list = filter (\x -> show x `isSuffixOf` show (x^2)) a008851_list
-- Reinhard Zumkeller, Jul 27 2011

[n: n in [0..10^7] | Intseq(n^2)[1..#Intseq(n)] eq Intseq(n)]; // Vincenzo Librandi, Jul 03 2015

V:= proc(m) option remember;
  select(t -> t^2 - t mod 10^m = 0, map(s -> seq(10^(m-1)*j+s, j=0..9), V(m-1)))
end proc:
V(0):= {0,1}:
V(1):= {5,6}:
sort(map(op,[V(0),seq(V(i) minus V(i-1),i=1..50)])); # Robert Israel, Jun 19 2015

f[k_] := (r = Reduce[0 < 10^k < n < 10^(k + 1) && n^2 == m*10^(k + 1) + n, {n, m}, Integers]; If[Head[r] === And, n /. ToRules[r], n /. {ToRules[r]}]); Flatten[ Join[{0, 1}, Table[f[k], {k, 0, 13}]]] (* Jean-François Alcover, Dec 01 2011 *)
Union@ Join[{1}, Array[PowerMod[5, 2^#, 10^#] &, 16, 0], Array[PowerMod[16, 5^#, 10^#] &, 16, 0]] (* Robert G. Wilson v, Jul 23 2018 *)

is_A003226(n)={n<2 || 10^valuation(n^2-n,10)>n} \\ M. F. Hasler, Dec 05 2012

A003226(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 } \\ M. F. Hasler, Dec 06 2012

from itertools import count, islice
from sympy.ntheory.modular import crt
def A003226_gen(): # generator of terms
    a = 0
    yield from (0,1)
    for n in count(0):
        b = sorted((int(crt(m:=(1< a:
            yield from b
            a = b[1]
        elif b[1] > a:
            yield b[1]
            a = b[1]
A003226_list = list(islice(A003226_gen(),15)) # Chai Wah Wu, Jul 25 2022

def automorphic(maxdigits, pow, base=10) :
    morphs = [[0]]
    for i in range(maxdigits):
        T=[d*base^i+x for x in morphs[-1] for d in range(base)]
        morphs.append([x for x in T if x^pow % base^(i+1) == x])
    res = list(set(sum(morphs, []))); res.sort()
    return res
# call with pow=2 for this sequence, Eric M. Schmidt, Feb 09 2014

Equals {0, 1} union A007185 union A016090.

More terms from Michel ten Voorde, Apr 11 2001
Edited by David W. Wilson, Sep 26 2002
Incorrect statement removed from title by Robert Dawson, Jul 09 2018

A317905 Convergence speed of m^^m, where m = A067251(n) and n >= 2. a(n) = f(m, m) - f(m, m - 1), where f(x, y) corresponds to the maximum value of k, such that x^^y == x^^(y + 1) (mod 10^k).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 6, 1, 1, 3, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1

Offset: 2

Marco Ripà, Aug 10 2018

It is possible to anticipate the convergence speed of m^^m, where ^^ indicates tetration or hyper-4 (e.g., 3^^4 = 3^(3^(3^3))), simply looking at the congruence (mod 25) of m. In fact, assuming m > 2, a(n) = 1 for any m == 2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23 (mod 25), and a(n) >= 2 otherwise.
It follows that 32/45 = 71.11% of the a(n) assume unitary value.
You can also obtain an arbitrary high convergence speed, such as taking the beautiful base b = 999...99 (9_9_9... n times), which gives a(n) = len(b), for any len(b) > 1. Thus, 99...9^^m == 99...9^^(m + 1) (mod m*10^len(b)), as proved by Ripà in "La strana coda della serie n^n^...^n", pages 25-26. In fact, m = 99...9 == 24 (mod 25) and a(m=24) > 1.
From Marco Ripà, Dec 19 2021: (Start)
Knowing the "constant congruence speed" of a given base (a.k.a. the convergence speed of the base m, assuming m > 2) is very useful in order to calculate the exact number of stable digits of all its tetrations of height b > 1. As an example, let us consider all the a(n) such that n is congruent to 4 (mod 9) (i.e., all the tetration bases belonging to the congruence class 5 (mod 10)). Then, the exact number of stable digits (#S(m, b)) of any tetration m^^b (i.e., the number of its last "frozen" digits) such that m is congruent to 5 (mod 10), for any b >= 3, can automatically be calculated by simply knowing that (under the stated constraint) the congruence speed of the m corresponds to the 2-adic valuation of (m^2 - 1) minus 1. Thus, let k = 1, 2, 3, ..., and we have that
If m = 20*k - 5, then #S(m, b > 2) = b*(v_2(m^2 - 1) - 1) + 1 = b*(v_2(m + 1) + 1);
If m = 20*k + 5, then #S(m, b > 2) = (b + 1)*(v_2(m^2 - 1) - 1) = (b + 1)*(v_2(m - 1));
If m = 5, then #S(m, 1) = 1, #S(m, 2) = 4, #S(m, b > 2) = 8 + 2*(b - 3).
(End)
For any n > 2, the value of a(n) depends on the congruence modulo 18 of n, since the constant congruence speed of m arises from the 14 nontrivial solutions of the fundamental equation y^5 = y in the (commutative) ring of decadic integers (e.g., y = -1 = ...9999 is a solution of y^5 = y, so it originates the law a(n) = min(v_2(m + 1), v_5(m + 1)) concerning every n belonging to the congruence class 0 modulo 18, as stated in the "Formula" section of the present sequence). - Marco Ripà, Feb 17 2022
a(n) satisfies the following multiplicative constraint: for each pair (m_1, m_2) of terms of A067251, a(m_1*m_2) is necessarily greater than or equal to the minimum between a(m_1) and a(m_2) (see Equation 2.4 and Appendix of "A Compact Notation for Peculiar Properties Characterizing Integer Tetration" in Links). - Gabriele Di Pietro, Apr 29 2025

			For m = 25, a(23) = 3 implies that 25^^(25 + i) freezes 3*i "new" rightmost digits (i >= 0).

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

Michel Marcus, reducetower.gp (from Math StackExchange).
Math StackExchange, Calculating a^n (mod m) in the general case
Marco Ripà, On the Convergence Speed of Tetration, ResearchGate (2018).
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.
Marco Ripà and Gabriele Di Pietro, A Compact Notation for Peculiar Properties Characterizing Integer Tetration, Zenodo, 2025.
Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441-457.
Wikipedia, Tetration

Cf. A067251, A317824, A317903, A349425, A370211, A370775, A371129, A371671, A371720, A372490, A373387.

Cf. A000007, A018247, A018248, A063006, A091661, A091663, A091664, A120817, A120818, A290372, A290373, A290374, A290375.

\\ uses reducetower.gp from links
f2(x,y) = my(k=0); while(reducetower(x, 10^k, y) == reducetower(x, 10^k, y+1), k++); k;
f1(n) = polcoef(x*(x+1)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1) / ((x-1)^2*(x^2+x+1)*(x^6+x^3+1)) + O(x^(n+1)), n, x); \\ A067251
a(n) =  my(m=f1(n)); f2(m, m) - f2(m, m-1);
lista(nn) = {for (n=2, nn, print1(a(n), ", "););} \\ Michel Marcus, Jan 27 2021

Let n > 2. For any integer c >= 0, if n is an element of the set {5, 7, 14, 17, 22, 23, 24, 29, 32, 39, 41, 45, 46}, then a(n + 45*c) >= 2; whereas a(n) = 1 otherwise. - Marco Ripà, Sep 28 2018
If n == 5 (mod 9), then a(n) = v_2(a(n)^2 - 1) - 1, where v_2(x) indicates the 2-adic valuation of x. - Marco Ripà, Dec 19 2021
If n == 1 (mod 18) and n <> 1, then a(n) = min(v_2(m - 1), v_5(m - 1)) (i.e., 1 plus the number of trailing zeros, if any, next to the rightmost digit of m);
if n == 10 (mod 18), then a(n) = min(v_2(m + 1), v_5(m - 1));
if n == {2,8}(mod 9) and n <> 2, then a(n) = v_5(m^2 + 1);
if n == {3,7}(mod 18), then a(n) = min(v_2(m + 1), v_5(n^2 + 1));
if n == {12,16}(mod 18), then a(n) = min(v_2(m - 1), v_5(n^2 + 1));
if n == 4 (mod 9), then a(n) = v_5(m + 1);
if n == 5 (mod 18), then a(n) = v_2(m - 1);
if n == 14 (mod 18), then a(n) = v_2(m + 1);
if n == 6 (mod 9), then a(n) = v_5(m - 1);
if n == 9 (mod 18), then a(n) = min(v_2(m - 1), v_5(m + 1));
if n == 0 (mod 18), then a(n) = min(v_2(m + 1), v_5(m + 1)) (i.e., number of digits of the rightmost repunit "9's" of m); where v_2(x) and v_5(x) indicates the 2-adic valuation of (x) and the 5-adic valuation of (x), respectively. - Marco Ripà, Feb 17 2022

Edited by Jinyuan Wang, Aug 30 2020

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

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

A091664 10-adic integer x=.....06619977392256259918212890624 satisfying x^3 = x.

Original entry on oeis.org

4, 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

Offset: 0

Edoardo Gueglio (egueglio(AT)yahoo.it), Jan 28 2004

Let a,b be integers defined in A018247, A018248 satisfying a^2=a, b^2=b, obviously a^3=a, b^3=b; let c,d,e,f be integers defined in A091661, A063006, A091663, A091664; then c^3=c, d^3=d, e^3=e, f^3=f, c+d=1, a+e=1, b+f=1, b+c=a, d+f=e, a+f=c, a=f+1, b=e+1, cd=-1, af=-1, gh=-1 where -1=.....999999999.

			x equals the limit of the (n+1) trailing digits of 4^(5^n):
4^(5^0) = (4), 4^(5^1) = 10(24), 4^(5^2) = 1125899906842(624), ...
x = ...0557423423230896109004106619977392256259918212890624.

Seiichi Manyama, Table of n, a(n) for n = 0..9999 (terms 0..999 from Paul D. Hanna)

Cf. A018247, A018248, A120817, A120818, A091661, A063006, A091663.

To calculate c, d, e, f use Mathematica algorithms for a, b and equations: c=a-b, d=1-c, e=b-1, f=a-1.

{a(n)=local(b=4,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

(A091664_vec(n)=Vecrev(digits(lift(chinese(Mod(0,2^n),Mod(-1,5^n))))))(99) \\ M. F. Hasler, Jan 26 2020

x = r^2 where r = ...1441224165530407839804103263499879186432 (A120817). x = 10-adic lim_{n->oo} 4^(5^n). - Paul D. Hanna, Jul 06 2006

For n > 0, a(n) = 9 - A018248(n) = A018247(n). - Seiichi Manyama, Jul 28 2017

A091663 10-adic integer x=.....93380022607743740081787109375 satisfying x^3 = x.

Original entry on oeis.org

5, 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

Offset: 0

Edoardo Gueglio (egueglio(AT)yahoo.it), Jan 28 2004

Let a,b be integers defined in A018247, A018248 satisfying a^2=a, b^2=b, obviously a^3=a, b^3=b; let c,d,e,f be integers defined in A091661, A063006, A091663, A091664 then c^3=c, d^3=d, e^3=e, f^3=f, c+d=1, a+e=1, b+f=1, b+c=a, d+f=e, a+f=c, a=f+1, b=e+1, cd=-1, af=-1, gh=-1 where -1=.....999999999.

Seiichi Manyama, Table of n, a(n) for n = 0..9999

Cf. A018247, A018248.

To calculate c, d, e, f use Mathematica algorithms for a, b and equations: c=a-b, d=1-c, e=b-1, f=a-1.

For n > 0, a(n) = 9 - A018247(n) = A018248(n). - Seiichi Manyama, Jul 28 2017

cp's OEIS Frontend

A120817 10-adic integer x=...07839804103263499879186432 satisfying x^5 = x; also x^3 = -x = A120818; (x^2)^3 = x^2 = A091664; (x^4)^2 = x^4 = A018248.

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A120818 10-adic integer x=...92160195896736500120813568 satisfying x^5 = x; also x^3 = -x = A120817; (x^2)^3 = x^2 = A091664; (x^4)^2 = x^4 = A018248.

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A075693 Difference between 10-adic numbers defined in A018248 & A018247.

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A076308 Product of 10-adic numbers defined in A018247 and A018248.

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A003226 Automorphic numbers: m^2 ends with m.

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A317905 Convergence speed of m^^m, where m = A067251(n) and n >= 2. a(n) = f(m, m) - f(m, m - 1), where f(x, y) corresponds to the maximum value of k, such that x^^y == x^^(y + 1) (mod 10^k).

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

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