A033819 -id:A033819 - cp's OEIS frontend

A003226 Automorphic numbers: m^2 ends with m.

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

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

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

A224473 (2*5^(2^n) - 1) mod 10^n: a sequence of trimorphic numbers ending in 9.

Original entry on oeis.org

9, 49, 249, 1249, 81249, 781249, 5781249, 25781249, 425781249, 6425781249, 36425781249, 836425781249, 9836425781249, 19836425781249, 519836425781249, 2519836425781249, 12519836425781249, 512519836425781249, 4512519836425781249, 84512519836425781249

Offset: 1

Eric M. Schmidt, Apr 07 2013

a(n) is the unique positive integer less than 10^n such that a(n) - 1 is divisible by 2^n and a(n) + 1 is divisible by 5^n.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Trimorphic Number
Index entries for sequences related to automorphic numbers

Cf. A033819. Corresponding 10-adic number is A091661. The other trimorphic numbers ending in 9 are included in A002283, A198971 and A224475.

def A224473(n) : return crt(1, -1, 2^n, 5^n);

a(n) = (2 * A007185(n) - 1) mod 10^n.

A068407 Automorphic numbers: numbers k such that k^5 ends with k. Also m-morphic numbers for all m > 5 such that m-1 is not divisible by 10 and m == 1 (mod 4).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 25, 32, 43, 49, 51, 57, 68, 75, 76, 93, 99, 125, 193, 249, 251, 307, 375, 376, 432, 443, 499, 501, 557, 568, 624, 625, 693, 749, 751, 807, 875, 943, 999, 1249, 1251, 1693, 1875, 2057, 2499, 2501, 2943, 3125, 3307, 3568, 3749, 3751

Offset: 1

Benoit Cloitre, Mar 08 2002

			13568 is a term because 13568^5 = 459810807237016813568 ends with 13568.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Index entries for sequences related to automorphic numbers

Cf. A003226, A033819.

[n : n in [0..3749] | Intseq(n^5)[1..#Intseq(n)] eq Intseq(n)]; // Arkadiusz Wesolowski, Nov 15 2013

Select[Range[0,100000],PowerMod[#,5,10^IntegerLength[#]]==#&] (* Harvey P. Dale, Nov 04 2011 *)

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])
    return sorted(set(sum(morphs,[])))
# (call with pow=5 for this sequence), Eric M. Schmidt, Jul 29 2013

A224474 (2*16^(5^n) - 1) mod 10^n: a sequence of trimorphic numbers ending in 1.

Original entry on oeis.org

1, 51, 751, 8751, 18751, 218751, 4218751, 74218751, 574218751, 3574218751, 63574218751, 163574218751, 163574218751, 80163574218751, 480163574218751, 7480163574218751, 87480163574218751, 487480163574218751, 5487480163574218751, 15487480163574218751

Offset: 1

Eric M. Schmidt, Apr 07 2013

a(n) is the unique positive integer less than 10^n such that a(n) + 1 is divisible by 2^n and a(n) - 1 is divisible by 5^n.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Trimorphic Number
Index entries for sequences related to automorphic numbers

Cf. A033819. Corresponding 10-adic number is A063006. The other trimorphic numbers ending in 1 are included in A199685 and A224476.

def A224474(n) : return crt(-1, 1, 2^n, 5^n)

a(n) = (2 * A016090(n) - 1) mod 10^n.

A288845 Values of n such that 4^n ends in n, or expomorphic numbers in base 4.

Original entry on oeis.org

6, 96, 896, 8896, 28896, 728896, 1728896, 11728896, 411728896, 90411728896, 290411728896, 5290411728896, 55290411728896, 555290411728896, 2555290411728896, 302555290411728896, 2302555290411728896, 22302555290411728896, 622302555290411728896, 3622302555290411728896

Offset: 1

Bernard Schott, Jun 18 2017

Definition: For positive integers b (as base) and n, the positive integer (allowing initial zeros) a(n) is expomorphic relative to base b (here 4) 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.]

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, "0411728896" would be included because the last 10 digits of 4^0411728896 are 0411728896, and also 02555290411728896" because the last 17 digits of 4^02555290411728896 are "02555290411728896". However, these are not in the sequence as defined here. - Jon E. Schoenfield

			4^6 = 4096 ends in 6, so 6 is a term; 4^96 = ....896 ends in 96, so 96 is another term.

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

Cf. A064541 (base 2), A183613 (base 3).

Cf. A003226 (automorphic numbers), A033819 (trimorphic numbers), A133614.

isok(n) = 10^valuation(4^n-n, 10)>n; \\ after A003226; Michel Marcus, Jun 18 2017

is(n)=my(m=10^#digits(n)); Mod(4, m)^n==n \\ Charles R Greathouse IV, Aug 10 2017

a(6)-a(9) from Gheorghe Coserea, Jun 21 2017

a(10)-a(11) from Robert G. Wilson v, Jun 24 2017

A306686 Values of n such that 9^n ends in n, or expomorphic numbers relative to "base" 9.

Original entry on oeis.org

9, 89, 289, 5289, 45289, 745289, 2745289, 92745289, 392745289, 7392745289, 97392745289, 597392745289, 7597392745289, 87597392745289, 8087597392745289, 48087597392745289, 748087597392745289, 10748087597392745289, 610748087597392745289, 5610748087597392745289

Offset: 1

Bernard Schott, Mar 05 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 9) 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.
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 A133619(0) through A133619(n-1). So, a(1) = 9, a(2) = 89 and so on, with recognition of the former comments about the OEIS and terms beginning with 0. - Davis Smith, Mar 07 2019

			9^9 = 387420489 ends in 9, so 9 is a term; 9^89 = .....289 ends in 89, so 89 is another term.

Davis Smith, Table of n, a(n) for n = 1..944
Charles W. Trigg, Problem 559, Crux Mathematicorum, page 192, Vol. 7, Jun. 81.
Emil Vaughan, Problem 226.8 - 999 nines, M500 Magazine of the Open University, number 226, February 2009, page 21; and Tony Forbes, Solution 226.8 - 999 nines, M500 Magazine of the Open University, number 232, February 2010, pages 8-9, calculating a(9) = 392745289.

Cf. A064541 (base 2), A183613 (base 3), A288845 (base 4), A306570 (base 5), A290788 (base 6), A321970 (base 7), A289138 (smallest expomorphic number in base n).
Cf. A003226 (automorphic numbers), A033819 (trimorphic numbers).
Cf. A133619 (leading digits).

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(9,n,10^n)!=tetrmod(9,n-1,10^(n-1)),print1(tetrmod(9,n,10^(n-1)),", "))) \\ Davis Smith, Mar 09 2019

a(8)-a(20) from Davis Smith, Mar 07 2019

A068408 Automorphic numbers: numbers k such that k^6 ends with k. Also m-morphic numbers for all m not congruent to 26 (mod 50) but congruent to 6 (mod 10).

Original entry on oeis.org

0, 1, 5, 6, 16, 21, 25, 36, 41, 56, 61, 76, 81, 96, 176, 201, 376, 401, 576, 601, 625, 776, 801, 976, 1376, 2001, 3376, 4001, 5376, 6001, 7376, 8001, 9376, 20001, 29376, 40001, 49376, 60001, 69376, 80001, 89376, 90625, 109376, 200001, 309376, 400001, 509376

Offset: 1

Benoit Cloitre, Mar 08 2002

90625^6 = 553972386755049228668212890625 hence 90625 is in the sequence.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Index entries for sequences related to automorphic numbers

Cf. A033819.

okQ[n_]:=Module[{idn=IntegerDigits[n],id6n=IntegerDigits[n^6]}, idn==Take[id6n,-Length[idn]]]
Select[Range[120000],okQ]  (* Harvey P. Dale, Jan 16 2011 *)

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=6 for this sequence), Eric M. Schmidt, Jul 29 2013

More terms from Eric M. Schmidt, Jul 29 2013

cp's OEIS Frontend

A003226 Automorphic numbers: m^2 ends with m.

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

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

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A224473 (2*5^(2^n) - 1) mod 10^n: a sequence of trimorphic numbers ending in 9.

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A068407 Automorphic numbers: numbers k such that k^5 ends with k. Also m-morphic numbers for all m > 5 such that m-1 is not divisible by 10 and m == 1 (mod 4).

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A224474 (2*16^(5^n) - 1) mod 10^n: a sequence of trimorphic numbers ending in 1.

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