A003226 -id:A003226 - cp's OEIS frontend

A056032 Trimorphic but not bimorphic nor automorphic.

4, 9, 24, 49, 51, 75, 99, 125, 249, 251, 375, 499, 501, 624, 749, 751, 875, 999, 1249, 3751, 4375, 4999, 5001, 5625, 6249, 8751, 9375, 9999, 18751, 31249, 40625, 49999, 50001, 59375, 68751, 81249, 90624, 99999, 109375, 218751, 281249, 390625, 499999

Offset: 1

Robert G. Wilson v, Jul 24 2000

Cf. A003226 and A033819.

Do[x=Floor[N[Log[10, n],25]]+1; If[Mod[n^3, 10^x] == n,If[Mod[n^2, 10^x]!= n, Print[n]]], {n,5*10^5}]

Missing a(41) inserted by Sean A. Irvine, Apr 12 2022

A056033 5-morphic but not bimorphic nor automorphic.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 24, 32, 43, 49, 51, 57, 68, 75, 93, 99, 125, 193, 249, 251, 307, 375, 432, 443, 499, 501, 557, 568, 624, 693, 749, 751, 807, 875, 943, 999, 1249, 1251, 1693, 1875, 2057, 2499, 2501, 2943, 3125, 3307, 3568, 3749, 3751, 4193, 4375, 4557, 4999

Offset: 1

Robert G. Wilson v, Jul 24 2000

Cf. A003226 and A033819.

Do[x=Floor[N[Log[10, n],25]]+1; If[Mod[n^5, 10^x] == n,If[Mod[n^2, 10^x]!= n, Print[n]]], {n, 10^4}]

A056036 5-morphic but not bimorphic, automorphic nor trimorphic.

Original entry on oeis.org

2, 3, 7, 8, 32, 43, 57, 68, 93, 193, 307, 432, 443, 557, 568, 693, 807, 943, 1251, 1693, 1875, 2057, 2499, 2501, 2943, 3125, 3307, 3568, 3749, 4193, 4557, 5443, 5807, 6251, 6432, 6693, 6875, 7057, 7499, 7501, 7943, 8125, 8307, 8749, 9193, 9557, 13568

Offset: 1

Robert G. Wilson v, Jul 24 2000

Cf. A003226 and A033819.

Do[x=Floor[N[Log[10, n],25]]+1; If[Mod[n^5, 10^x] == n,If[Mod[n^2, 10^x]!= n, If[Mod[n^3, 10^x]!= n,Print[n]]]], {n,1,50000}]

A056050 Semimorphic numbers: n^2 mod the number of digits in n = n^3 mod the number of digits in n.

Original entry on oeis.org

1, 5, 6, 10, 20, 25, 26, 30, 40, 45, 50, 60, 65, 70, 76, 80, 85, 90, 100, 200, 225, 300, 376, 400, 425, 500, 600, 625, 700, 800, 825, 876, 900, 1000, 1025, 1100, 1200, 1300, 1400, 1425, 1500, 1600, 1700, 1800, 1825, 1876, 1900, 2000, 2100, 2200, 2225, 2300

Offset: 1

Robert G. Wilson v, Jul 25 2000

Cf. A003226.

Do[l = Floor[N[Log[10, n], 24]] + 1; If[Mod[n^2, 10^l] == Mod[n^3, 10^l], Print[n]], {n, 1, 5000}]

A094190 Least n-digit automorphic number.

Original entry on oeis.org

0, 25, 376, 9376, 90625, 109376, 2890625, 12890625, 212890625, 1787109376, 18212890625, 918212890625, 9918212890625, 40081787109376, 259918212890625, 3740081787109376, 43740081787109376, 256259918212890625, 2256259918212890625

Offset: 1

Lekraj Beedassy, May 25 2004

Cf. A003226, A003226, A007185, A016090.

f[n_] := If[n == 1, 0, Block[{a5 = PowerMod[5, 2^n, 10^n], a6 = PowerMod[16, 5^n, 10^n]}, If[a5 < 10^(n - 1), a5 = 10^(n + 1)]; If[a6 < 10^(n - 1), a6 = 10^(n + 1)]; Min[a5, a6]]]; Array[f, 19] (* Robert G. Wilson v, Aug 27 2006 *)

a(n) = Min( 5^(2^n) (mod 10^n), 16^(5^n) (mod 10^n) ). - Robert G. Wilson v, Aug 27 2006

Corrected and extended by Robert G. Wilson v, Aug 27 2006

A177192 Primes p such that p^p ends in p and p is not congruent to 1 (mod 10).

Original entry on oeis.org

5, 193, 499, 557, 1249, 1693, 4999, 7057, 31249, 49999, 52057, 54193, 56249, 79193, 281249, 829193, 952057, 4531249, 4999999, 8281249, 8704193, 17077057, 39954193, 54577057, 63281249, 64954193, 904577057, 2154577057, 3092077057, 3958704193

Offset: 1

Jonathan Vos Post and Robert G. Wilson v, May 04 2010

A proper subset of A052228.

Eric Weisstein's World of Mathematics, Automorphic Number.
Index entries for sequences related to automorphic numbers
Index entries for sequences related to final digits of numbers

Cf. A003226, A007185, A016090, A052228.

fQ[n_] := PowerMod[n, n, 10^Floor[Log[10, n] + 1]] == n; p = 2; lst = {}; While[p < 10^12, If[ Mod[p, 10] != 1 && fQ@p, AppendTo[lst, p]; Print@p]; p = NextPrime@p]; lst

A308249 Squares of automorphic numbers in base 12 (cf. A201918).

Original entry on oeis.org

0, 1, 16, 81, 4096, 6561, 263169, 1478656, 40960000, 205549569, 54988374016, 233605955584, 6263292059649, 303894740860929, 338531738189824, 170196776412774400, 709858175909625856, 18638643564726714369, 124592287100855910400, 2576097707358918017025, 479214351668445504864256

Offset: 1

Jeremias M. Gomes, May 17 2019

(sqrt(m))12 is a suffix of m_12. - _A.H.M. Smeets, Aug 09 2019

All terms k^2 in this sequence (except the trivials 0 and 1) have a square root k that is the suffix of one of the 12-adic numbers given by A259468 or A259469. From this, the sequence has an infinite number of terms. - A.H.M. Smeets, Aug 09 2019

			4096 = 2454_12 and sqrt(2454_12) = 54_12. Hence 4096 is in the sequence.

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

Cf. A201918, A259468, A259469.

dig = "0123456789AB"
def To12(n):
    s = ""
    while n > 0:
        s, n = dig[n%12]+s, n//12
    return s
n, m = 1, 0
print(n,m*m)
while n < 100:
    m = m+1
    m2, m1 = To12(m*m), To12(m)
    i, i2, i1 = 0, len(m2), len(m1)
    while i < i1 and (m2[i2-i-1] == m1[i1-i-1]):
        i = i+1
    if i == i1:
        print(n,m*m)
n = n+1 # A.H.M. Smeets, Aug 09 2019

[(n * n) for n in (0..1000000) if (n * n).str(base = 12).endswith(n.str(base = 12))]

Equals A201918(n)^2.

Terms a(16)..a(21) from A.H.M. Smeets, Aug 09 2019

A351410 Numbers m such that the decimal representation of 8^m ends in m.

Original entry on oeis.org

56, 856, 5856, 25856, 225856, 5225856, 95225856, 895225856, 6895225856, 16895225856, 416895225856, 5416895225856, 35416895225856, 7035416895225856, 77035416895225856, 577035416895225856, 1577035416895225856, 21577035416895225856, 521577035416895225856, 1521577035416895225856, 81521577035416895225856

Offset: 1

Bernard Schott, Feb 10 2022

The Crux Mathematicorum link calls these numbers "expomorphic" relative to "base" b, with here b = 8.
Under that definition, the term after a(13) = 35416895225856 is not "035416895225856" or "35416895225856" but a(14) = 7035416895225856.
Conjecture: if k(n) is "expomorphic" relative to "base" b, then the next one in the sequence, k(n+1), consists of the last n+1 digits of b^k(n).
This conjecture is true. See A133618. - David A. Corneth, Feb 10 2022

			8^56 = 374144419156711147060143317175368453031918731001856, so 56 is a term.
8^856 = ...5856 ends in 856, so 856 is another term.

Charles W. Trigg, Problem 559, Crux Mathematicorum, pp. 192-194, Vol. 7, Jun. 1981.

Cf. A064541, A183613, A288845, A306570, A290788, A321970, this sequence, A306686, A289138.
Cf. A003226 (automorphic numbers), A033819 (trimorphic numbers).
Cf. A133618 (leading digits).

a(7)-a(8) from Michel Marcus, Feb 10 2022

More terms from David A. Corneth, Feb 10 2022

A385171 Perfect powers m^k whose decimal expansion begins with k and ends with m, where m and k are greater than 1.

Original entry on oeis.org

25, 59049, 78125, 13060694016, 17179869184, 19073486328125, 30514648531249, 53613724194557, 59120987373568, 65944160601201, 116490258898219, 324965351768751, 512908935546875, 21936950640377856, 371308922853718751, 578261433548013568, 913517247483640899

Offset: 1

Gonzalo Martínez, Jun 20 2025

Such as automorphic numbers (A003226), which are those m such that m^2 ends with m, if m^k is in this sequence, then it is a k-morphic number which also begins with k. Thus, m^k contains both m and k as substrings at its ends.
If m is in A003226 and m^2 starts with 2, then m^2 is in this sequence. For example, A003226(3)^2 = 5^2 = 25 and A003226(119)^2.
If m is in A033819 and m^3 starts with 3, then m^3 is in this sequence. For example, A033819(39)^3 = 31249^3 = 30514648531249.
This sequence has infinitely many terms since (10^m - 1)^9 is a term for all m >= 2, which starts with (m - 1) 9's and ends with m 9's.

			6^13 = 13060694016 is a term since it starts with 13 and ends with 6.

Cf. A001597, A003226, A033819, A378857, A051248, A074250.

cp's OEIS Frontend

A056032 Trimorphic but not bimorphic nor automorphic.

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A056033 5-morphic but not bimorphic nor automorphic.

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A056036 5-morphic but not bimorphic, automorphic nor trimorphic.

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A056050 Semimorphic numbers: n^2 mod the number of digits in n = n^3 mod the number of digits in n.

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Mathematica

A094190 Least n-digit automorphic number.

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A177192 Primes p such that p^p ends in p and p is not congruent to 1 (mod 10).

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A308249 Squares of automorphic numbers in base 12 (cf. A201918).

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A351410 Numbers m such that the decimal representation of 8^m ends in m.

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A385171 Perfect powers m^k whose decimal expansion begins with k and ends with m, where m and k are greater than 1.

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