A003226 -id:A003226 - cp's OEIS frontend

A052228 Automorphic primes: p such that p^p ends with the digits of p.

5, 11, 31, 41, 61, 71, 101, 151, 193, 251, 401, 499, 557, 601, 701, 751, 1151, 1201, 1249, 1301, 1601, 1693, 1801, 1901, 1951, 2351, 2551, 2801, 3001, 3301, 3701, 4001, 4201, 4751, 4801, 4951, 4999, 5101, 5351, 5501, 5701, 5801, 6101, 6151, 6301, 6551

Offset: 1

G. L. Honaker, Jr., Jan 30 2000

			11 is in the sequence because 11 is prime and 11^11=285311670611 ends in 11.

Harvey P. Dale, Table of n, a(n) for n = 1..500

Cf. A003226.

Select[ Prime@ Range@863, PowerMod[ #, #, 10^Floor[Log[10, # ] + 1]] == # &] (* Robert G. Wilson v Sep 25 2006 *)
Select[Prime[Range[850]],PowerMod[#,#,10^IntegerLength[#]]==#&] (* Harvey P. Dale, Mar 05 2015 *)

More terms from Jason Earls, Jan 02 2002

A290788 Values of n such that 6^n ends in n, or expomorphic numbers in "base" 6.

Original entry on oeis.org

6, 56, 656, 8656, 38656, 238656, 7238656, 47238656, 447238656, 7447238656, 27447238656, 227447238656, 3227447238656

Offset: 1

Bernard Schott, Aug 10 2017

Definition: For positive integers b (as base) and n, the positive integer (allowing initial 0's) a(n) is expomorphic relative to base b (here 6) 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.]

			6^6 = 46656 ends in 6, so 6 is a term.
6^56 = ...656 ends in 56, so 56 is another term.

Charles W. Trigg, Problem 559, Crux Mathematicorum, page 192, Vol. 7, Jun. 81.

Cf. A064541 (base 2), A183613 (base 3), A288845 (base 4), A289138, A306570 (base 5), A306686 (base 9).

Cf. A003226 (automorphic numbers), A033819 (trimorphic numbers).

Select[Range[10^6], PowerMod[6, #, 10^(1 + Floor@ Log10[#])] == # &] (* Michael De Vlieger, Apr 13 2021 *)

is(n)=my(m=10^#digits(n)); Mod(6,m)^n==n \\ Charles R Greathouse IV, Aug 10 2017

a(6)-a(9) from Charles R Greathouse IV, Aug 10 2017

a(10)-a(13) from Chai Wah Wu, Apr 13 2021

A067270 Numbers m such that m-th triangular number (A000217) ends in m.

Original entry on oeis.org

0, 1, 5, 25, 625, 9376, 90625, 890625, 7109376, 12890625, 212890625, 1787109376, 81787109376, 59918212890625, 259918212890625, 3740081787109376, 56259918212890625, 256259918212890625, 7743740081787109376

Offset: 1

Joseph L. Pe, Feb 21 2002

Thanks to David W. Wilson for the proof that this sequence is a proper subset of A003226.
Also, numbers m such that the m-th k-gonal number ends in m for k == 1, 3, 5, or 9 (mod 10). - Robert Dawson, Jul 09 2018
This sequence is the intersection of A093534 and A301912. - Robert Dawson, Aug 01 2018

			The 5th triangular = 15 ends in 5, hence 5 is a term of the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..888
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.

Proper subset of A003226. Cf. A007185, A018247, A016090, A018248.

Intersection of A093534 and A301912.

(* a5=A018247 less the commas; a6=A018248 less the commas; *)
b5 = FromDigits[ Reverse[ IntegerDigits[a5]]]; b6 = FromDigits[ Reverse[ IntegerDigits[a6]]]; f[0] = 1; f[n_] := Block[{c5 = Mod[b5, 10^n], c6 = Mod[b6, 10^n]}, If[ Mod[c5(c5 + 1)/2, 10^n] == c5, c5, c6]]; Union[ Table[ f[n], {n, 0, 20}]]

from itertools import count, islice
from sympy.ntheory.modular import crt
def A067270_gen(): # generator of terms
    a = 0
    yield from (0,1)
    for n in count(0):
        if (b := int(min(crt(m:=(1<<(n+1),5**n),(0,1))[0], crt(m,(1,0))[0]))) > a:
            yield b
            a = b
A067270_list = list(islice(A067270_gen(),15)) # Chai Wah Wu, Jul 25 2022

Edited and extended by Robert G. Wilson v, Nov 20 2002

0 prepended by David A. Corneth, Aug 02 2018

A227071 Let s(m) = the set of k > 0 such that k^m ends with k. Then a(n) = least m such that s(m) = s(n).

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 3, 2, 9, 2, 11, 2, 5, 2, 3, 6, 17, 2, 3, 2, 21, 2, 3, 2, 9, 26, 3, 2, 5, 2, 11, 2, 33, 2, 3, 6, 5, 2, 3, 2, 41, 2, 3, 2, 5, 6, 3, 2, 17, 2, 51, 2, 5, 2, 3, 6, 9, 2, 3, 2, 21, 2, 3, 2, 65, 6, 3, 2, 5, 2, 11, 2, 9, 2, 3, 26, 5, 2, 3, 2, 81, 2

Offset: 1

T. D. Noe, Jul 29 2013

See A227070 for more details and for the numbers n such that n = a(n).

The entries in the b-file have been tentatively obtained by comparing the terms < 10^30 in the sets s(n). - Giovanni Resta, Jul 30 2013

Giovanni Resta, Table of n, a(n) for n = 1..10000 (warning: contains tentative terms)

Cf. A003226 (n=2), A033819 (n=3), A068407 (n=5), A068408 (n=6).
Cf. A072496 (n=11), A072495 (n=21), A076650 (n=26).
Cf. A227070 (n such that n = a(n)).

ts = {{}}; t2 = {1}; te = {1}; Do[s = Select[Range[0, 10^7], PowerMod[#, n, 10^IntegerLength[#]] == # &]; If[MemberQ[ts, s], AppendTo[t2, te[[Position[ts, s, 1, 1][[1, 1]]]]], AppendTo[ts, s]; AppendTo[te, n]; AppendTo[t2, n]], {n, 2, 82}]; t2

Conjecture: a(n+1) = A132741(n) + 1. - Eric M. Schmidt, Jul 30 2013

Mathematica program and some entries corrected by Giovanni Resta, Jul 30 2013

A269588 Numbers n such that n^2 ends with the digits of n reversed (A004086(n)).

Original entry on oeis.org

1, 5, 6, 963, 9867, 65766, 69714, 6317056, 90899553, 169605719, 4270981082, 96528287587, 465454256742, 692153612536, 182921919071841, 655785969669834, 650700037578750084, 125631041500927357539, 673774165549097456624, 16719041449406813636569

Offset: 1

José Eduardo Gaboardi de Carvalho, Mar 01 2016

a(29)>10^32 (if it exists)

			6317056^2 = 39905196507136 which ends with 6507136, so 6317056 is a term.

Robert Gerbicz, Table of n, a(n) for n = 1..28
IBM, "Ponder This" puzzle for March 2016

Subsequence of A115761.

Cf. A003226, A004086.

Select[Range[10^7], Function[k, Take[IntegerDigits[#^2], -Length@ k] == Reverse@ k]@ IntegerDigits@ # &] (* Michael De Vlieger, Mar 04 2016 *)

isA269588(n)=dn = digits(n); rn = subst(Polrev(dn), x, 10); nbd = #dn; (n^2 - rn) % 10^nbd == 0; \\ Michel Marcus, Mar 01 2016

\\ printA269588len(d) prints all terms of the sequence with d digits
rev(n) = eval(concat(Vecrev(Str(n))));
{ printA269588len(d) = my(l, u, n); l=ceil(d/2); u=floor(d/2); for(y=0, 10^l-1, n=rev(y^2 % 10^u)*10^l+y; if(#Str(n)==d && Mod(n, 10^d)^2==rev(n), print(n)); ); }
\\ Max Alekseyev, Mar 07 2016

a(18)-a(20) from Max Alekseyev, Mar 07 2016
a(21)-a(27) from Robert Gerbicz, Apr 03 2016
a(28) from Dieter Beckerle, Jun 09 2016

A094534 Centered hexamorphic numbers: the k-th centered hexagonal number, 3k(k-1)+1, ends in k.

Original entry on oeis.org

1, 7, 17, 51, 67, 167, 251, 417, 501, 667, 751, 917, 1251, 1667, 5001, 5417, 6251, 6667, 10417, 16667, 50001, 56251, 60417, 66667, 166667, 260417, 406251, 500001, 666667, 760417, 906251, 1406251, 1666667, 5000001, 5260417, 6406251, 6666667, 16666667

Offset: 1

Robert Munafo, May 07 2004

Given any number in the sequence, if you remove one or more digits from the beginning you always get another number in the sequence. This makes it easy to find higher terms -- just take an existing term and try adding a digit (with perhaps additional 0's) at the beginning. For example, to 6251 prepend 5 to get a 5-digit term, or 40 or 90 to get a 6-digit term.

			417 is in the sequence because if n=417, 3n(n-1)+1=520417, which ends in 417.

Robert Munafo, Sequence A094534, Centered Hexamorphic, or Automorphic Hexagonal, Numbers
Cliff Pickover, Centered Hexamorphic Numbers.

Cf. A003215, A003226.

isok(n) = {my(m = 3*n*(n-1)+1); (m - n) % 10^#Str(n) == 0; } \\ Michel Marcus, Jun 21 2018

10^(d-1) <= n < 10^d; 3n(n-1)+1 == n mod 10^d

Name changed by Robert Dawson, Jun 20 2018

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

A270343 Numbers k that end with ( sum of digits of k )^2.

Original entry on oeis.org

0, 1, 81, 3144, 3256, 6225, 6484, 6576, 7121, 7529, 7676, 9100, 9324, 9361, 9729, 9784, 12144, 12256, 15225, 15484, 15576, 16121, 16529, 16676, 18100, 18324, 18361, 18729, 18784, 21144, 21256, 24225, 24484, 24576, 25121

Offset: 1

Soumil Mandal, Mar 15 2016

All terms end with a digit from the set S = {0,1,4,5,6,9}.

The sum of the digits of the numbers repeat and also change with regular intervals. For example, the sum of the digits S1 = {12,16,15,22,24,11,23,26,10,18,19,27,28} which is followed by 3144 to 8784, 12144 to 18784, 21144 to 27784, 30144 to 36784. Again S2 = {21,25,15,22,24,11,23,26,10,18,19,27,28} is followed by 39441 to 45784, 48441 to 54784, 57441 to 67784, 66441 to 72784. It can be seen that a set containing 13 elements repeats itself for 4 consecutive ranges.

			For k=3256, sum of digits is 16 and 16^2 is 256.
For k=7121, sum of digits is 11 and 11^2 is 121.
For k=18784, sum of digits is 22 and 22^2 is 484.

Paolo P. Lava, Table of n, a(n) for n = 1..10000
Soumil Mandal, Graph Of Sum Of Digits
Soumil Mandal, Graph Of Cumulative Sums

Cf. A003226, A008851, A018247, A118881.

Select[Range[0, 20000], Function[n, Function[k, If[n >= k, FromDigits@ Take[#, -IntegerLength@ k] == k, False]][Total[#]^2] &@ IntegerDigits@ n]] (* Michael De Vlieger, Mar 15 2016 *)
esdQ[n_]:=Module[{idn=IntegerDigits[n],idn2=IntegerDigits[ Total[ IntegerDigits[ n]]^2]},Take[ idn,-Length[idn2]]==idn2]; Select[ Range[ 0,26000],esdQ]//Quiet (* Harvey P. Dale, Jan 01 2022 *)

isok(n) = {sds = sumdigits(n)^2; nbs = #Str(sds); ((n - sds) % 10^nbs) == 0;} \\ Michel Marcus, Mar 16 2016

for i in range(0,200000):
    res = pow((sum(map(int,str(i)))),2)
    if(i%pow(10,len(str(res)))==res):print(i)
# Soumil Mandal, Mar 17 2016

A308262 Numbers m such that A048385(m) ends with m.

Original entry on oeis.org

0, 1, 5, 6, 10, 11, 25, 36, 50, 51, 60, 61, 100, 101, 110, 111, 250, 251, 360, 361, 425, 500, 501, 510, 511, 600, 601, 610, 611, 936, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 1936, 2500, 2501, 2510, 2511, 3600, 3601, 3610, 3611, 4250, 4251, 5000, 5001

Offset: 1

Rémy Sigrist, May 17 2019

If m belongs to this sequence, then A048385(m) belongs to this sequence.
If m belongs to this sequence, then 10*m and 10*m + 1 belong to this sequence.
This sequence contains A007088.
All terms belong to A052419.
Let U be the infinite word ...|A048385^2(16)|A048385(16)|16425 and V be the infinite word ...|A048385^2(81)|A048385(81)|81936. The terms of this sequence consist of the last x digits of either U or V followed by y digits in {0,1}, where x and y are nonnegative integers. - Charlie Neder, May 17 2019

			The first terms, alongside A048385(a(n)), are:
  n   a(n)  A048385(a(n))
  --  ----  -------------
   1     0              0
   2     1              1
   3     5             25
   4     6             36
   5    10             10
   6    11             11
   7    25            425
   8    36            936
   9    50            250
  10    51            251
  11    60            360
  12    61            361

Rémy Sigrist, PARI program for A308262

Cf. A003226, A007088, A048385, A052419.

m=1;
for u=0:5001
    digit=dec2base(u,10)-'0';digitp=digit.^2;
    aa=str2num(strrep(num2str(digitp), ' ', ''));
    digitaa=dec2base(aa,10)-'0';
       if mod(aa,10^length(digit))==u
        sol(m)=u; m=m+1;
       end
end
sol % Marius A. Burtea, May 17 2019

PARI
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See Links section.
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A383821 3-automorphic numbers: positive integers k such that 3k^2 ends with k.

Original entry on oeis.org

2, 5, 7, 67, 75, 92, 667, 792, 875, 6667, 6875, 9792, 66667, 69792, 96875, 296875, 369792, 666667, 2369792, 4296875, 6666667, 62369792, 66666667, 262369792, 404296875, 666666667, 6666666667, 7262369792, 9404296875, 27262369792, 39404296875, 66666666667, 639404296875

Offset: 1

Shyam Sunder Gupta, May 11 2025

All 3-automorphic numbers end in 2, 5, or 7 only.
From Michael S. Branicky, May 11 2025: (Start)
Terms of successively larger digits can be created by prepending digits on the left of previous terms; for each length, only 3 positive such "seeds" are valid (some may have leading zeros and thus do not contribute terms at that length).
Infinite since 6..67, with i 6's and then a 7 is a term for all i >= 0.
a(2774) has 1001 digits. (End)

			67 is in the sequence because 3*67^2 = 13467 which ends with 67.

Michael S. Branicky, Table of n, a(n) for n = 1..2773
Shyam Sunder Gupta, Elegance of Squares, Cubes, and Higher Powers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 2, 29-81.

Cf. A003226, A033428.

Essentially the union of A030985, A030986, and A067275.

Select[Range[10^7],IntegerDigits[#]==Take[IntegerDigits[3#^2],-IntegerLength[#]]&] (* James C. McMahon, May 16 2025 *)

More terms from Michael S. Branicky, May 11 2025

cp's OEIS Frontend

A052228 Automorphic primes: p such that p^p ends with the digits of p.

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A290788 Values of n such that 6^n ends in n, or expomorphic numbers in "base" 6.

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Examples

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Programs

Mathematica

PARI

Extensions

A067270 Numbers m such that m-th triangular number (A000217) ends in m.

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A227071 Let s(m) = the set of k > 0 such that k^m ends with k. Then a(n) = least m such that s(m) = s(n).

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A269588 Numbers n such that n^2 ends with the digits of n reversed (A004086(n)).

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PARI

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A094534 Centered hexamorphic numbers: the k-th centered hexagonal number, 3k(k-1)+1, ends in k.

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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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