A029943 -id:A029943 - 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

A018834 Numbers k such that the decimal expansion of k^2 contains k as a substring.

Original entry on oeis.org

0, 1, 5, 6, 10, 25, 50, 60, 76, 100, 250, 376, 500, 600, 625, 760, 1000, 2500, 3760, 3792, 5000, 6000, 6250, 7600, 9376, 10000, 14651, 25000, 37600, 50000, 60000, 62500, 76000, 90625, 93760, 100000, 109376, 250000, 376000, 495475, 500000, 505025

Offset: 1

David W. Wilson

			25^2 = 625 which contains 25.
3792^2 = 14_3792_64, 14651^2 = 2_14651_801.

Giovanni Resta, Table of n, a(n) for n = 1..200 (first 126 terms from David W. Wilson)

Cf. A003226, A046831, A046851, A045953.
Cf. A000290. Supersequence of A029943.
Cf. A018826 (base 2), A018827 (base 3), A018828 (base 4), A018829 (base 5), A018830 (base 6), A018831 (base 7), A018832 (base 8), A018833 (base 9).
Cf. A029942 (cubes), A075904 (4th powers), A075905 (5th powers).

import Data.List (isInfixOf)
a018834 n = a018834_list !! (n-1)
a018834_list = filter (\x -> show x `isInfixOf` show (x^2)) [0..]
-- Reinhard Zumkeller, Jul 27 2011

Select[Range[510000], MemberQ[FromDigits /@ Partition[IntegerDigits[#^2], IntegerLength[#], 1], #] &] (* Jayanta Basu, Jun 29 2013 *)
Select[Range[0,510000],StringPosition[ToString[#^2],ToString[#]]!={}&] (* Ivan N. Ianakiev, Oct 02 2016 *)

from itertools import count, islice
def A018834_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:str(n) in str(n**2), count(max(startvalue,0)))
A018834_list = list(islice(A018834_gen(),20)) # Chai Wah Wu, Apr 04 2023

A029942 Numbers k such that the decimal expansion of k^3 contains k as a substring.

Original entry on oeis.org

0, 1, 4, 5, 6, 9, 10, 24, 25, 32, 40, 49, 50, 51, 56, 60, 75, 76, 90, 99, 100, 125, 240, 249, 250, 251, 375, 376, 400, 490, 499, 500, 501, 510, 600, 624, 625, 749, 750, 751, 760, 782, 875, 900, 990, 999, 1000, 1249, 1250, 2400, 2490, 2500, 2510

Offset: 1

Patrick De Geest

			24 is a term as 24^3 = 13824 contains 24 as a substring.
250 is a term as 250^3 = 1562500 contains 250 as a substring.
6^3 = 21_6, 782^3 = 4_782_11768.

Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 500 terms from Reinhard Zumkeller)

Cf. A018834 (squares), A075904 (4th powers), A075905 (5th powers), A136490 (base 2).

Cf. A000578. Supersequence of A029943.

import Data.List (isInfixOf)
a029942 n = a029942_list !! (n-1)
a029942_list = [x | x <- [0..], show x `isInfixOf` show (x^3)]
-- Reinhard Zumkeller, Feb 29 2012

n3ssQ[n_]:=Module[{idn=IntegerDigits[n],idn3=Partition[ IntegerDigits[ n^3], IntegerLength[n],1]},MemberQ[idn3,idn]]; Join[{0},Select[Range[ 2600],n3ssQ]] (* Harvey P. Dale, Jan 23 2012 *)
Select[Range[0,2600],SequenceCount[IntegerDigits[#^3],IntegerDigits[ #]]> 0&] (* Harvey P. Dale, Aug 29 2021 *)

A178983 The smallest cube containing n as a substring.

Original entry on oeis.org

0, 1, 27, 343, 64, 125, 64, 27, 8, 729, 1000, 91125, 125, 1331, 140608, 15625, 216, 1728, 85184, 2197, 205379, 216, 226981, 103823, 13824, 125, 9261, 27, 1728, 729, 39304, 1331, 5832, 1331, 343, 35937, 97336, 3375, 13824, 39304, 4096, 531441, 42875, 343

Offset: 0

Andy Martin, Jan 02 2011

			a(3) = 7^3 = 343, because it contains 3 as a substring and no smaller cube contains 3.

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A029943, A029947.

subs[n_] := Module[{d = IntegerDigits[n], len}, len = Length[d]; Union[Flatten[Table[FromDigits[Take[d, {i, k}]], {k, len}, {i, k}]]]]; Table[k = 0; While[! MemberQ[subs[k^3], n], k++]; k^3, {n, 0, 100}] (* T. D. Noe, Nov 06 2013 *)
With[{cbs=Range[0,100]^3},Table[SelectFirst[cbs,SequenceCount[IntegerDigits[#],IntegerDigits[n]]>0&],{n,0,50}]] (* Harvey P. Dale, Nov 17 2024 *)

# For a given nonnegative integer n,
# find the smallest nonnegative cube that contains it as a substring.
NUM_TERMS = 30
(0...NUM_TERMS).each{ |i|
  (0..(1.0/0.0)).each{ |j|
    (print "#{j*j*j}" + ", "; break) if "#{j*j*j}".include?("#{i}")
  }
}

Edited by Alois P. Heinz, Jan 02 2011

A133408 Numbers k such that k is a substring of both its square and its cube in base 2 (written in base 10).

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 32, 41, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296

Offset: 1

Jonathan Vos Post, Dec 22 2007

Binary analog of A029943. Subset of A018826.
Row 2 of array whose row 1 is A002275 and whose row 10 is A029943.
Contains every power of 2. Is 41 the only term which is not a power of 2? - Sean A. Irvine, Oct 11 2009
Up to 1.7*10^13 the sequence does not contain numbers greater than 41 which are not a power of 2. - Giovanni Resta, Aug 30 2018

			41 is a term because 41 (base 2) = 101001, which is a substring of 41^2 (base 2) = 11010010001 and which is a substring of 41^3 (base 2) = 10000110100111001.

Cf. A007088, A018826, A029943.

Select[Range[0,10^6],SequenceCount[IntegerDigits[#^2,2],IntegerDigits[#,2]]>0&&SequenceCount[IntegerDigits[#^3,2],IntegerDigits[#,2]]>0&] (* James C. McMahon, Mar 17 2025 *)

{k such that A007088(k) is a substring of A007088(k^2) and is a substring of A007088(k^3)}.

Corrected and extended by Sean A. Irvine, Oct 11 2009
a(32) from Oliver Allen, Aug 08 2017
a(33)-a(35) from Oliver Allen, Aug 10 2017

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A003226 Automorphic numbers: m^2 ends with m.

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A018834 Numbers k such that the decimal expansion of k^2 contains k as a substring.

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A029942 Numbers k such that the decimal expansion of k^3 contains k as a substring.

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A178983 The smallest cube containing n as a substring.

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A133408 Numbers k such that k is a substring of both its square and its cube in base 2 (written in base 10).

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