A033819 -id:A033819 - cp's OEIS frontend

A074194 Trimorphic numbers: a(n)^3 == a(n) (mod 10^35).

0, 1, 9004106619977392256259918212890624, 9004106619977392256259918212890625, 18008213239954784512519836425781249, 31991786760045215487480163574218751, 40995893380022607743740081787109375, 49999999999999999999999999999999999, 50000000000000000000000000000000001, 59004106619977392256259918212890625, 68008213239954784512519836425781249, 81991786760045215487480163574218751, 90995893380022607743740081787109375, 90995893380022607743740081787109376, 99999999999999999999999999999999999

Offset: 1

Zak Seidov, Sep 19 2002

There are 15 trimorphs mod 10^n for n >= 3.

Index entries for sequences related to automorphic numbers

Cf. A033819.

Edited by David W. Wilson, Sep 26 2002

A074755 Number of n-digit non-leading-zero trimorphic numbers (m such that m^3 ends in m).

Original entry on oeis.org

5, 7, 13, 11, 11, 13, 12, 13, 12, 13, 13, 11, 10, 12, 12, 13, 12, 12, 12, 11, 13, 12, 12, 10, 10, 13, 13, 13, 11, 13, 12, 10, 10, 11, 11, 13, 13, 10, 13, 11, 13, 12, 13, 12, 12, 13, 12, 12, 12, 12, 12, 11, 10, 10, 13, 13, 13, 13, 13, 13, 11, 13, 12, 10, 11, 11, 13

Offset: 1

David W. Wilson, Sep 28 2002

For n >= 3, I can show 8 <= a(n) <= 13 and I highly suspect that 10 <= a(n) <= 13 from empirical evidence.

If n >= 3, there are 15 integers 0 <= x < 10^n with x == 0, 1 or -1 mod 5^n and x == 0, 1, -1, 2^(n-1)-1 or 2^(n-1)+1 mod 2^n, and a(n) is the number of these that are >= 10^(n-1). - Robert Israel, Jul 07 2014

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

Cf. A033819.

f:= n ->
  nops(select(`>=`,{seq(seq(chrem([a,b],[2^n,5^n]),a={0,1,2^(n-1)-1,2^(n-1)+1,-1}),b={0,1,-1})},10^(n-1))):
seq(f(n), n=1..100); # Robert Israel, Jul 07 2014

A175394 Least nontrivial exponent e > 2 such that n^2 is a substring of n^e (n >= 0).

Original entry on oeis.org

3, 3, 6, 6, 7, 3, 7, 6, 22, 11, 3, 13, 26, 54, 123, 27, 27, 40, 100, 43, 6, 43, 54, 42, 12, 3, 37, 43, 9, 37, 6, 19, 102, 102, 43, 96, 83, 45, 67, 34, 12, 128, 168, 102, 182, 44, 152, 104, 184, 52, 3, 17, 35, 75, 164, 67, 127, 22, 134, 98, 7, 124, 117, 146, 77, 146, 156, 87

Offset: 0

Zak Seidov, Apr 29 2010

			a(2)=6 because 2^2=4 is a substring of 2^6=64
a(4)=7 because 4^2=16 is a substring of 4^7=16384.

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A008856, A033819, A045537, A074250.

f:= proc(n) local pat,e;
  pat:= sprintf("%d",n^2);
  for e from 3 do if StringTools:-Search(pat, sprintf("%d",n^e))<> 0 then return e fi od:
end proc:
map(f, [$0..100]); # Robert Israel, Mar 20 2018

lne[n_]:=Module[{e=3,idn2=IntegerDigits[n^2]},While[!MemberQ[ Partition[ IntegerDigits[n^e], Length[ idn2],1],idn2],e++];e]; Array[lne,70,0] (* Harvey P. Dale, Aug 17 2013 *)

A215558 Cubes whose final digits are the cube root.

Original entry on oeis.org

0, 1, 64, 125, 216, 729, 13824, 15625, 117649, 132651, 421875, 438976, 970299, 1953125, 15438249, 15813251, 52734375, 53157376, 124251499, 125751501, 242970624, 244140625, 420189749, 423564751, 669921875, 997002999, 1948441249, 52776573751, 83740234375

Offset: 1

Jonathan Vos Post, Aug 16 2012

			6 * 6 * 6 = 216, which has "6" as rightmost digit.

Philip Carter & Ken Russell, Problem 31, p. 86, The Complete Book of Fun Maths, John Wiley, 2004.

Cf. A000578, A033819.

Select[Range[0,100]^3,Mod[#,10]==Surd[#,3]||Mod[#,100]==Surd[#,3]&] (* Harvey P. Dale, Mar 03 2023 *)

apply(x->x^3,select(x -> (x^3 - x) % 10^(#digits(x)) == 0, [0..99])) \\ David A. Corneth, Feb 11 2020

a(n) = A033819(n)^3.

A227070 Powers n such that the set s(n) = {k > 0 such that k^n ends with k} does not occur for smaller n.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 11, 17, 21, 26, 33, 41, 51, 65, 81, 101, 126, 129, 161, 201, 251, 257, 321, 401, 501, 513, 626, 641, 801, 1001, 1025, 1251, 1281, 1601, 2001, 2049, 2501, 2561, 3126, 3201, 4001, 4097, 5001, 5121, 6251, 6401, 8001, 8193, 10001

Offset: 1

T. D. Noe, Jul 29 2013

These numbers might be called automorphic powers because the sets s(n) are called automorphic numbers. It appears that all numbers of the form 1 + 5^i are here. In fact, these appear to produce the only even numbers here. The set s(4) equals s(2). The set s(7) equals s(3). The set s(9) does not differ from s(5) until k = 10443. The set s(17) does not differ from s(9) until k = 108307. The sequence also has 126, 201, 251, 501, and 626, but there may be missing numbers.

Entries a(17)-a(49) have been tentatively obtained by comparing the terms < 10^30 in the sets s(n), for 2 <= n <= 10001. - Giovanni Resta, Jul 30 2013

Cf. A003226 (n=2), A033819 (n=3), A068407 (n=5), A068408 (n=6).
Cf. A072496 (n=11), A072495 (n=21), A076650 (n=26).
Cf. A227071.

ts = {}; t = {}; Do[s = Select[Range[11000000], PowerMod[#, n, 10^IntegerLength[#]] == # &]; If[! MemberQ[ts, s], Print[n]; AppendTo[ts, s]; AppendTo[t, n]], {n, 2, 101}]; t = Join[{1}, t]

Conjecture: a(n+1) = A003592(n) + 1. - Eric M. Schmidt, Jul 30 2013

a(17)-a(49) from Giovanni Resta, Jul 30 2013

A056032 Trimorphic but not bimorphic nor automorphic.

Original entry on oeis.org

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}]

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

A074194 Trimorphic numbers: a(n)^3 == a(n) (mod 10^35).

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A074755 Number of n-digit non-leading-zero trimorphic numbers (m such that m^3 ends in m).

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A175394 Least nontrivial exponent e > 2 such that n^2 is a substring of n^e (n >= 0).

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A215558 Cubes whose final digits are the cube root.

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A227070 Powers n such that the set s(n) = {k > 0 such that k^n ends with k} does not occur for smaller n.

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

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