login
This site is supported by donations to The OEIS Foundation.

 

Logo

The submissions stack has been unacceptably high for several months now. Please voluntarily restrict your submissions and please help with the editing. (We don't want to have to impose further limits.)

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A003226 Automorphic numbers: n^2 ends with n. Also m-morphic numbers for all m not equal to 6 (mod 10).
(Formerly M3752)
30
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

REFERENCES

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 253-4.

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).

Xiaolong Ron Yu, Curious Numbers, Pi Mu Epsilon Journal, Spring 1999, pp. 819-823.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

W. Schneider, Automorphic Numbers

Eric Weisstein's World of Mathematics, Automorphic Number

Wikipedia, Automorphic number

Index entries for sequences related to automorphic numbers

FORMULA

Equals {0, 1} union A007185 union A016090.

Let b(n)=Sum_{0<= x <= floor(log[10](n))} ((floor(n^2/10^x)-10*floor(n^2/10^(x + 1)))*10^x)- n. For n >= 1, b(n)=0 if and only if n is an automorphic number. - José de Jesús Camacho Medina, Jun 17 2015

MAPLE

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

MATHEMATICA

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 *)

a = Table[Sum[(Floor[n^2/10^x] - 10*Floor[n^2/10^(x + 1)])*10^x, {x, 0, Floor[Log[10, n]] }] , {n, 1, 9400}] - Table[n, {n, 1, 9400}]

Position[A, 0]

(* José de Jesús Camacho Medina, Jun 17 2015 *)

PROG

(Haskell)

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

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

(PARI) 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

(Sage)

def automorphic(maxdigits, pow, base=10) :

....morphs = [[0]]

....for i in xrange(maxdigits):

........T=[d*base^i+x for x in morphs[-1] for d in xrange(base)]

........morphs.append([x for x in T if x^pow % base^(i+1) == x])

....res = list(set(sum(morphs, []))); res.sort()

....return res

end # (call with pow=2 for this sequence), Eric M. Schmidt, Feb 09 2014

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

CROSSREFS

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

Sequence in context: A136888 A038248 A046831 * A088834 A137081 A137079

Adjacent sequences:  A003223 A003224 A003225 * A003227 A003228 A003229

KEYWORD

nonn,base,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Michel ten Voorde (seqfan(AT)tenvoorde.org), Apr 11 2001

Edited by David W. Wilson, Sep 26 2002

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified August 28 03:16 EDT 2015. Contains 261112 sequences.