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

Author

N. J. A. Sloane

Keywords

• nonn
• base
• nice
• easy

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

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

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 253-254.
• 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).

Links

• 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

Crossrefs

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

Programs

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

Magma

[n: n in [0..10^7] | Intseq(n^2)[1..#Intseq(n)] eq Intseq(n)]; // Vincenzo Librandi, Jul 03 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 *)
Union@ Join[{1}, Array[PowerMod[5, 2^#, 10^#] &, 16, 0], Array[PowerMod[16, 5^#, 10^#] &, 16, 0]] (* Robert G. Wilson v, Jul 23 2018 *)

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

Python

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

Sage

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

Formula

Equals {0, 1} union A007185 union A016090.

Extensions

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