A003226 -id:A003226 - cp's OEIS frontend

A201821 Automorphic numbers: n^2 ends with n in base 6 (written in base 6).

0, 1, 3, 4, 13, 44, 213, 344, 5344, 50213, 205344, 350213, 1350213, 4205344, 21350213, 34205344, 221350213, 334205344, 2221350213, 3334205344, 52221350213, 152221350213, 403334205344, 5152221350213, 55152221350213, 155152221350213, 400403334205344

Offset: 1

Martin Renner, Dec 06 2011

			a(3) = (3)_6 = 3 since 3^2 = 9 = (13)_6 ends with 3 in base 6.
a(4) = (4)_6 = 4 since 4^2 = 16 = (24)_6 ends with 4 in base 6.
a(5) = (13)_6 = 9 since 9^2 = 81 = (213)_6 ends with 13 in base 6.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 147.
Maurice Kraitchik, Mathematical Recreations, New York, Dover, (2nd ed.) 1953, p. 77.

Cf. A237583 (written in base 10), A003226, A201918, A201919, A201921, A201948.

More terms from Eric M. Schmidt, Feb 09 2014

A201919 Automorphic numbers n^2 ends with n in base 14 (written in base 10).

Original entry on oeis.org

0, 1, 7, 8, 49, 148, 344, 2401, 36016, 151264, 386561, 1764736, 5764801, 46941952, 58471553, 374712065, 1101076992, 4802079233, 15858967552, 139825248256, 149429406721, 1595702681601, 2453862488064, 14602557997056, 42091354378241, 127990382747648

Offset: 1

Martin Renner, Dec 06 2011

			a(3) = 7 = (7)_14 since 7^2 = 49 = (37)_14 ends with 7 in base 14.
a(4) = 8 = (8)_14 since 8^2 = 64 = (48)_14 ends with 8 in base 14.
a(5) = 49 = (37)_14 since 49^2 = 2401 = (C37)_14 ends with 37 in base 14.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A003226, A201821, A201918, A201921, A201948, A237583.

# See A003226. - Eric M. Schmidt, Feb 09 2014

More terms from Eric M. Schmidt, Feb 09 2014

A046831 Numbers k such that decimal expansion of k^2 contains k as a substring and k does not end in 0.

Original entry on oeis.org

1, 5, 6, 25, 76, 376, 625, 3792, 9376, 14651, 90625, 109376, 495475, 505025, 890625, 971582, 1713526, 2890625, 4115964, 5133355, 6933808, 7109376, 10050125, 12890625, 48588526, 50050025, 66952741, 87109376, 88027284, 88819024

Offset: 1

David W. Wilson

Subsequence of A018834. - Chai Wah Wu, Apr 04 2023

Giovanni Resta, Table of n, a(n) for n = 1..64

Cf. A018834, A003226, A008851.

a046831 n = a046831_list !! (n-1)
a046831_list = filter ((> 0) . (`mod` 10)) a018834_list
-- Reinhard Zumkeller, Jul 27 2011

Reap[For[n = 1, n < 10^8, n++, If[Mod[n, 10] != 0, If[StringPosition[ToString[n^2], ToString[n]] != {}, Print[n]; Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Apr 04 2013 *)

from itertools import count, islice
def A046831_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:n%10 and str(n) in str(n**2), count(max(startvalue,0)))
A046831_list = list(islice(A046831_gen(),20)) # Chai Wah Wu, Apr 04 2023

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

A055620 Digits of an idempotent 6-adic number.

Original entry on oeis.org

4, 4, 3, 5, 0, 2, 4, 3, 3, 3, 0, 4, 0, 0, 4, 1, 4, 2, 4, 3, 0, 0, 0, 5, 0, 3, 0, 0, 0, 2, 4, 1, 2, 2, 5, 1, 3, 3, 1, 5, 4, 2, 2, 4, 1, 5, 3, 5, 4, 3, 0, 3, 1, 5, 3, 2, 2, 5, 2, 1, 0, 0, 3, 0, 0, 1, 2, 3, 2, 4, 0, 1, 0, 1, 5, 4, 4, 5, 1, 3, 5, 4, 2, 5, 4, 0, 5, 1, 2, 0, 5, 4, 5, 3, 1, 5, 2, 1, 3, 3, 2, 3, 3, 5, 3

Offset: 0

Paolo Dominici (pl.dm(AT)libero.it), Jun 04 2000

( a(0) + a(1)*6 + a(2)*6^2 + ... )^k = a(0) + a(1)*6 + a(2)*6^2 + ... for each k. Apart from 0 and 1 in base 6 there are only 2 numbers with this property. For the other see A054869.

			(a(0) + a(1)*6 + a(2)*6^2 + a(3)*6^3)^2 == (a(0) + a(1)*6 + a(2)*6^2 + a(3)*6^3) mod 6^4 because 1478656 == 1216 (mod 1296).

V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.

Kenny Lau, Table of n, a(n) for n = 0..9999
V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.

The six examples given by deGuerre and Fairbairn are A055620, A054869, A018247, A018248, A259468, A259469.

first(p)=Vecrev(digits(lift(Mod(4,6^p)^3^p), 6)) \\ Charles R Greathouse IV, Nov 01 2022

n=10000;res=pow((3**n+1)//2,n,3**n)*2**n
for i in range(n):print(i,res%6);res//=6
# Kenny Lau, Jun 09 2018

If A is the 6-adic number, A == 4^(3^n) mod 6^n. - Robert Dawson, Oct 28 2022

A201918 Automorphic numbers: n^2 ends with n in base 12 (written in base 10).

Original entry on oeis.org

0, 1, 4, 9, 64, 81, 513, 1216, 6400, 14337, 234496, 483328, 2502657, 17432577, 18399232, 412549120, 842530816, 4317249537, 11162091520, 50755272705, 692253097984, 2178269839360, 6737830608897, 46758772080640, 60234433298433, 474731593596928, 809186870951937

Offset: 1

Martin Renner, Dec 06 2011

			a(3) = 4 = (4)_12 since 4^2 = 16 = (14)_12 ends with 4 in base 12.
a(4) = 9 = (9)_12 since 9^2 = 81 = (69)_12 ends with 9 in base 12.
a(5) = 64 = (54)_12 since 64^2 = 4096 = (2454)_12 ends with 54 in base 12.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A003226, A201821, A201919, A201921, A201948, A237583.

a201918[n_Integer] := Module[{i = 0}, Flatten[Last[Reap[
     Do[If[
       IntegerDigits[i^2, 12][[-Length[IntegerDigits[i, 12]] ;; -1]] ==
         IntegerDigits[i, 12], Sow[i]], {i, n}]]]]]; a201918[12^6] (* Michael De Vlieger, Aug 13 2014 *)

# See A003226. - Eric M. Schmidt, Feb 09 2014

More terms from Eric M. Schmidt, Feb 09 2014

A259468 Digits of an idempotent 12-adic number.

Original entry on oeis.org

4, 5, 8, 3, 11, 1, 6, 11, 1, 2, 11, 2, 5, 4, 3, 11, 7, 6, 2, 11, 6, 8, 9, 6, 1, 10, 3, 10, 1, 8, 10, 3, 3, 1, 10, 8, 6, 4, 6, 11, 2, 0, 4, 0, 3, 8, 9, 9, 5, 3, 1, 0, 9, 7, 11, 8, 7, 6, 5, 2, 5, 1, 10, 2, 11, 0, 2, 7, 6, 6, 5, 4, 7, 2, 0, 5, 11, 7, 8, 7, 2, 6

Offset: 0

N. J. A. Sloane, Jul 02 2015

V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.

Eric M. Schmidt, Table of n, a(n) for n = 0..9999
V. deGuerre and R. A. Fairbairn, Automorphic numbers, Jnl. Rec. Math., 1 No. 3, (1968), 173-179.

The six examples given by deGuerre and Fairbairn are A055620, A054869, A018247, A018248, A259468, A259469.

def a_list(n) : return crt(1, 0, 3^n, 4^n).digits(12) # Eric M. Schmidt, Jul 09 2015

More terms from Eric M. Schmidt, Jul 09 2015

A259469 Digits of an idempotent 12-adic number.

Original entry on oeis.org

9, 6, 3, 8, 0, 10, 5, 0, 10, 9, 0, 9, 6, 7, 8, 0, 4, 5, 9, 0, 5, 3, 2, 5, 10, 1, 8, 1, 10, 3, 1, 8, 8, 10, 1, 3, 5, 7, 5, 0, 9, 11, 7, 11, 8, 3, 2, 2, 6, 8, 10, 11, 2, 4, 0, 3, 4, 5, 6, 9, 6, 10, 1, 9, 0, 11, 9, 4, 5, 5, 6, 7, 4, 9, 11, 6, 0, 4, 3, 4, 9, 5, 11

Offset: 0

N. J. A. Sloane, Jul 02 2015

V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.

Eric M. Schmidt, Table of n, a(n) for n = 0..9999
V. deGuerre and R. A. Fairbairn, Automorphic numbers, Jnl. Rec. Math., 1 (No. 3, 1968), 173-179.

The six examples given by deGuerre and Fairbairn are A055620, A054869, A018247, A018248, A259468, A259469.

def a_list(n) : return crt(0, 1, 3^n, 4^n).digits(12) # Eric M. Schmidt, Jul 09 2015

More terms from Eric M. Schmidt, Jul 09 2015

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

A029943 Substring of both its square and its cube.

Original entry on oeis.org

0, 1, 5, 6, 10, 25, 50, 60, 76, 100, 250, 376, 500, 600, 625, 760, 1000, 2500, 3760, 5000, 6000, 6250, 7600, 9376, 10000, 25000, 37600, 50000, 60000, 62500, 76000, 90625, 93760, 100000, 109376, 250000, 376000, 500000, 600000, 625000

Offset: 1

Patrick De Geest

Intersection of A018834 and A029942. - Reinhard Zumkeller, Feb 29 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Automorphic Number
Wikipedia, Automorphic number
Index entries for sequences related to automorphic numbers

import Data.List (isInfixOf)
a029943 n = a029943_list !! (n-1)
a029943_list = filter f [0..] where
   f x = show x `isInfixOf` show (x^2) && show x `isInfixOf` show (x^3)
-- Reinhard Zumkeller, Nov 26 2011

ssscQ[n_]:=Module[{idn=IntegerDigits[n],sq=IntegerDigits[n^2], cu=IntegerDigits[n^3],len=IntegerLength[n]},MemberQ[Partition[ sq,len,1], idn] &&MemberQ[Partition[cu,len,1],idn]]; Join[{0}, Select[Range[700000],ssscQ]] (* Harvey P. Dale, Apr 24 2011 *)

a(n) = A003226(m) * 10^k for appropriate m and k. [Reinhard Zumkeller, Nov 26 2011]

Offset corrected by Reinhard Zumkeller, Nov 26 2011

cp's OEIS Frontend

A201821 Automorphic numbers: n^2 ends with n in base 6 (written in base 6).

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A201919 Automorphic numbers n^2 ends with n in base 14 (written in base 10).

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A046831 Numbers k such that decimal expansion of k^2 contains k as a substring and k does not end in 0.

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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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A055620 Digits of an idempotent 6-adic number.

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PARI

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A201918 Automorphic numbers: n^2 ends with n in base 12 (written in base 10).

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A259468 Digits of an idempotent 12-adic number.

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Sage

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A259469 Digits of an idempotent 12-adic number.

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