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A003226 Automorphic numbers: n^2 ends with n. Also m-morphic numbers for all m not equal to 6 (mod 10).
(Formerly M3752)
29
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]

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.

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.

Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Hoboken, 2005, p. 64.

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.

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-Fran├žois Alcover, Dec 01 2011 *)

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

CROSSREFS

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

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

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Last modified October 31 05:50 EDT 2014. Contains 248845 sequences.