A016090 -id:A016090 - cp's OEIS frontend

A018248 The 10-adic integer x = ...1787109376 satisfies x^2 = x.

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

Offset: 0

Yoshihide Tamori (yo(AT)salk.edu)

The 10-adic numbers a and b defined in A018247 and this sequence satisfy a^2=a, b^2=b, a+b=1, ab=0. - Michael Somos

			x equals the limit of the (n+1) trailing digits of 6^(5^n):
6^(5^0)=(6), 6^(5^1)=77(76), 6^(5^2)=28430288029929701(376), ...
x = ...9442576576769103890995893380022607743740081787109376.
From _Peter Bala_, Nov 05 2022: (Start)
Trailing digits of 2^(10^n), 4^(10^n) and 6^(10^n) for n = 5:
2^(10^5) = ...9883(109376);
4^(10^5) = ...7979(109376);
6^(10^5) = ...4155(109376). (End)

W. W. R. Ball, Mathematical Recreations & Essays, N.Y. Macmillan Co, 1947.
R. Cuculière, Jeux Mathématiques, in Pour la Science, No. 6 (1986), 10-15.
V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.
M. Kraitchik, Sphinx, 1935, p. 1.
A. M. Robert, A Course in p-adic Analysis, Springer, 2000; see pp. 63, 419.

Seiichi Manyama, Table of n, a(n) for n = 0..9999 (terms 0..999 from Paul D. Hanna).
Anonymous, Automorphic numbers (2)
Peter Bala, A note on A018248
V. deGuerre and R. A. Fairbairn, Automorphic numbers, Jnl. Rec. Math., 1 (No. 3, 1968), 173-179.
MathOverflow, Distribution of digits of pq-adic idempotents (a.k.a. "automorphic numbers"), 2014.
Eric Weisstein's World of Mathematics, Automorphic numbers (1)
Index entries for sequences related to automorphic numbers

A016090 gives associated automorphic numbers.
Cf. A018247, A033819.
The difference between this sequence & A018247 is A075693 and their product is A075693.
Cf. A120817, A120818, A091664.
The six examples given by deGuerre and Fairbairn are A055620, A054869, A018247, A018248, A259468, A259469.

a := proc (n) option remember; if n = 1 then 2 else irem(a(n-1)^10, 10^n) end if; end proc:
# display the digits of a(100) from right to left
S := convert(a(100), string):
with(ListTools):
the_List := [seq(parse(S[i]), i = 1..length(S))]:
Reverse(the_List); # Peter Bala, Nov 04 2022

b = {6}; g[n_] := Block[{k = 0, c}, While[c = FromDigits[Prepend[b, k]]; Mod[c^2, 10^n] != c, k++ ]; b = Prepend[b, k]]; Do[ g[n], {n, 2, 105}]; Reverse[b]
With[{n = 150}, Reverse[IntegerDigits[PowerMod[16, 5^n, 10^n]]]] (* IWABUCHI Yu(u)ki, Feb 16 2024 *)

{a(n)=local(b=6,v=[]);for(k=1,n+1,b=b^5%10^k;v=concat(v,(10*b\10^k)));v[n+1]} \\ Paul D. Hanna, Jul 06 2006

Vecrev(digits(lift(chinese(Mod(0, 2^100), Mod(1, 5^100))))) \\ Seiichi Manyama, Aug 07 2019

x = r^4 where r=...1441224165530407839804103263499879186432 (A120817). x = 10-adic limit_{n->oo} 6^(5^n). - Paul D. Hanna, Jul 06 2006

For n >= 2, the final n+1 digits of either 2^(10^n), 4^(10^n) or 6^(10^n), when read from right to left, give the first n+1 entries in the sequence. - Peter Bala, Nov 05 2022

More terms from David W. Wilson

Edited by David W. Wilson, Sep 26 2002

A033819 Trimorphic numbers: n^3 ends with n. Also m-morphic numbers for all m > 5 such that m-1 is not divisible by 10 and m == 3 (mod 4).

Original entry on oeis.org

0, 1, 4, 5, 6, 9, 24, 25, 49, 51, 75, 76, 99, 125, 249, 251, 375, 376, 499, 501, 624, 625, 749, 751, 875, 999, 1249, 3751, 4375, 4999, 5001, 5625, 6249, 8751, 9375, 9376, 9999, 18751, 31249, 40625, 49999, 50001, 59375, 68751, 81249, 90624, 90625

Offset: 1

David W. Wilson

n is in this sequence iff it occurs in one of A002283, A007185, A016090, A198971, A199685, A216092, A216093, A224473, A224474, A224475, A224476, A224477, and A224478. - Eric M. Schmidt, Apr 08 2013

Let q(n) = floor(a(n)^3 / 10^A055642(a(n))), where A055642(n) is the number of digits in the decimal expansion of n. As well, let na and nb denote the indices of the preceding and next terms that begin with a 9. Then (1/q(n)) * (a(n)^4 - a(n)^3 - a(n)^2 + a(n)) - 2*a(n)^2 + a(n) + q(n) + 1 = a(na+nb-n)^2 - a(na+nb-n) - q(na+nb-n). - Christopher Hohl, Apr 08 2019

			376^3 = 53157376 which ends with 376.

S. Premchaud, A class of numbers, Math. Student, 48 (1980), 293-300.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
Shyam Sunder Gupta, Elegance of Squares, Cubes, and Higher Powers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 2, 29-81.
Eric Weisstein's World of Mathematics, Trimorphic Number
Index entries for sequences related to automorphic numbers

Cf. A074194, A215558 (cubes of the terms).

[n: n in [0..10^5] | Intseq(n^3)[1..#Intseq(n)] eq Intseq(n)]; // Bruno Berselli, Apr 04 2013

Do[x=Floor[N[Log[10, n], 25]]+1; If[Mod[n^3, 10^x] == n, Print[n]], {n, 1, 10000}]
Select[Range[100000],PowerMod[#,3,10^IntegerLength[#]]==#&](* Harvey P. Dale, Nov 04 2011 *)
Select[Range[0, 10^5], 10^IntegerExponent[#^3-#, 10]>#&] (* Jean-François Alcover, Apr 04 2013 *)

A216093 a(n) = 10^n - (5^(2^n) mod 10^n).

Original entry on oeis.org

5, 75, 375, 9375, 9375, 109375, 7109375, 87109375, 787109375, 1787109375, 81787109375, 81787109375, 81787109375, 40081787109375, 740081787109375, 3740081787109375, 43740081787109375, 743740081787109375

Offset: 1

V. Raman, Sep 01 2012

nonn

a(n)^3 mod 10^n = a(n).
a(n) is the unique positive integer less than 10^n such that a(n) is divisible by 5^n and a(n) + 1 is divisible by 2^n. - Eric M. Schmidt, Sep 01 2012
a(n+1) + a(n)^2 == 0 (mod 10^(n+1)). - Robert Israel, Apr 24 2017

Robert Israel, Table of n, a(n) for n = 1..999

Cf. A007185, A016090, A216092, A091663, A018248.

f:= n -> (-5 &^(2^n) mod 10^n):
map(f, [$1..30]); # Robert Israel, Apr 24 2017

def A216093(n) : return crt(-1, 0, 2^n, 5^n) # Eric M. Schmidt, Sep 01 2012

2^(4*5^(n-1)) mod 10^n - 1.

A224474 (2*16^(5^n) - 1) mod 10^n: a sequence of trimorphic numbers ending in 1.

Original entry on oeis.org

1, 51, 751, 8751, 18751, 218751, 4218751, 74218751, 574218751, 3574218751, 63574218751, 163574218751, 163574218751, 80163574218751, 480163574218751, 7480163574218751, 87480163574218751, 487480163574218751, 5487480163574218751, 15487480163574218751

Offset: 1

Eric M. Schmidt, Apr 07 2013

a(n) is the unique positive integer less than 10^n such that a(n) + 1 is divisible by 2^n and a(n) - 1 is divisible by 5^n.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Trimorphic Number
Index entries for sequences related to automorphic numbers

Cf. A033819. Corresponding 10-adic number is A063006. The other trimorphic numbers ending in 1 are included in A199685 and A224476.

def A224474(n) : return crt(-1, 1, 2^n, 5^n)

a(n) = (2 * A016090(n) - 1) mod 10^n.

A216092 a(n) = 2^(2*5^(n-1)) mod 10^n.

Original entry on oeis.org

4, 24, 624, 624, 90624, 890624, 2890624, 12890624, 212890624, 8212890624, 18212890624, 918212890624, 9918212890624, 59918212890624, 259918212890624, 6259918212890624, 56259918212890624, 256259918212890624

Offset: 1

V. Raman, Sep 01 2012

nonn

a(n) is the unique positive integer less than 10^n such that a(n) is divisible by 2^n and a(n) + 1 is divisible by 5^n. - Eric M. Schmidt, Sep 01 2012

Robert Israel, Table of n, a(n) for n = 1..996

Cf. A007185, A016090, A216093, A091664, A018247.

f:= n -> 2&^(2*5^(n-1)) mod 10^n:
map(f, [$1..100]); # Robert Israel, Mar 13 2025

Table[PowerMod[5,2^n,10^n],{n,20}]-1 (* Harvey P. Dale, Dec 17 2017 *)

def A216092(n) : return crt(0, -1, 2^n, 5^n) # Eric M. Schmidt, Sep 01 2012

a(n) = (5^(2^n) mod 10^n) - 1.
a(n)^3 == a(n) (mod 10^n).
a(n-1) == a(n) (mod 10^(n-1)). - Robert Israel, Mar 13 2025

A075693 Difference between 10-adic numbers defined in A018248 & A018247.

Original entry on oeis.org

1, 5, -3, 9, -9, -7, 5, 7, 5, -7, 7, -9, -9, -1, 5, -3, -1, 5, 5, -9, 3, -5, -5, -9, -9, 7, -3, -3, 9, 7, 1, 9, 9, -9, 9, 7, -3, -9, -7, 9, 3, 5, 3, 5, 1, 3, 5, 1, -5, -1, -1, 9, 9, 9, 7, 7, -7, 3, -3, -7, 9, -7, -1, -9, 9, -1, -3, -3, 7, 5, -3, 9, 9, -9, -7, -9, 9, -1, -7, 3, -9, 5, 9, -7

Offset: 0

Robert G. Wilson v, Sep 26 2002

Numbers in A018247 and A018248 are known as automorphic numbers in base 10, meaning that the infinite integers a=(...256259918212890625) or b=(...743740081787109376) provides a nontrivial solution to x*x == x (mod any power of 10).
Read backwards so as to match their counterparts (A007185 & A016090), A018247(0)+A018248(0) = 11 & A018247(n)+A018248(n) = 9 for all n's > 0 and their product is A076308.
All entries must be odd.
Is the accumulative sum equally positive and negative, i.e. does the sum equal 0 infinitely often?

Seiichi Manyama, Table of n, a(n) for n = 0..9999

Cf. A018248 & A018247.

(* execute the programming in both A018247 & A018248 *) Reverse[b - a]
aa[n_] := For[t = 5; k = 1, True, k++, t = Mod[t^2, 10^k]; If[k == n, Return[ Quotient[t, 10^(n-1)]]]]; bb[n_] := Reap[ For[t = 6; k = 1, k <= n , k++, t = Mod[t^5, 10^k]; Sow[ Quotient[10*t, 10^k]]]][[2, 1, n]]; a[n_] := bb[n] - aa[n]; Table[a[n], {n, 1, 84}](* Jean-François Alcover, May 25 2012, after Paul D. Hanna *)

a(n) = A018248(n) - A018247(n). - Seiichi Manyama, Jul 26 2017

A113627 a(n) is the smallest number k such that k and 2^k have the same last n digits. Here k may have fewer than n digits and can be padded with leading zeros (cf. A121319).

Original entry on oeis.org

14, 36, 736, 8736, 48736, 948736, 2948736, 32948736, 432948736, 3432948736, 53432948736, 353432948736, 5353432948736, 75353432948736, 75353432948736, 5075353432948736, 15075353432948736, 615075353432948736, 8615075353432948736, 98615075353432948736, 98615075353432948736

Offset: 1

Jon E. Schoenfield, Apr 23 2007

			2^14 = 16384 and 14 end with the same single digit 4, thus a(1) = 14.

Jon E. Schoenfield, Table of n, a(n) for n = 1..81

See A121319, the main entry for this sequence, for further information.
Same as A109405 except for the initial term (14). - Max Alekseyev, May 11 2007
Cf. A007185, A016090, A003226, A035383, A064540, A064541, A109405, A121319, A206636.

A121319 a(n) is the smallest number k such that k and 2^k have the same last n digits. Here k must have at least n digits (cf. A113627).

Original entry on oeis.org

14, 36, 736, 8736, 48736, 948736, 2948736, 32948736, 432948736, 3432948736, 53432948736, 353432948736, 5353432948736, 75353432948736, 1075353432948736, 5075353432948736, 15075353432948736, 615075353432948736, 8615075353432948736, 98615075353432948736

Offset: 1

Tanya Khovanova, Aug 25 2006

			2^14 = 16384 and 14 end with the same single digit 4, thus a(1) = 14.

Jon E. Schoenfield, Table of n, a(n) for n = 1..80
Jon E. Schoenfield, Excel program

Cf. A007185, A016090, A003226, A035383, A064540, A064541, A109405.

f[n_] := Block[{k = If[n == 1, 2, 10], m = 10^n}, While[ PowerMod[2, k, m] != Mod[k, m], k += 2]; k]; Do[ Print@f@n, {n, 9}] (* Robert G. Wilson v *)

A121319(n) = { local(k,tn); tn=10^n ; forstep(k=2,1000000000,2, if ( k % tn == (2^k) % tn, return(k) ; ) ; ) ; return(0) ; } { for(n = 1,13, print( A121319(n)) ; ) ; } \\ R. J. Mathar, Aug 27 2006

If A109405(n) has n digits, a(n) = A109405(n), otherwise a(n) = A109405(n) + 10^n. - Max Alekseyev, May 05 2007

a(6)-a(9) from Robert G. Wilson v and Jon E. Schoenfield, Aug 26 2006
a(10) from Robert G. Wilson v, Sep 26 2006
a(11)-a(16) from Alexander Adamchuk, Jan 28 2007
a(16) corrected by Max Alekseyev, Apr 12 2007

A224478 (16^(5^n) + (10^n)/2 - 1) mod 10^n: a sequence of trimorphic numbers ending (for n > 1) in 5.

Original entry on oeis.org

0, 25, 875, 4375, 59375, 609375, 2109375, 37109375, 287109375, 6787109375, 31787109375, 581787109375, 5081787109375, 90081787109375, 240081787109375, 8740081787109375, 93740081787109375, 243740081787109375, 2743740081787109375, 57743740081787109375

Offset: 1

Eric M. Schmidt, Apr 07 2013

a(n) is the unique nonnegative integer less than 10^n such that a(n) + 2^(n-1) + 1 is divisible by 2^n and a(n) is divisible by 5^n.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Trimorphic Number
Index entries for sequences related to automorphic numbers

Cf. A033819. Converges to the 10-adic number A091663. The other trimorphic numbers ending in 5 are included in A007185, A216093, and A224477.

def A224478(n) : return crt(2^(n-1)-1, 0, 2^n, 5^n)

a(n) = (A016090(n) + 10^n/2 - 1) mod 10^n.

A301912 Numbers k such that the decimal representation of k ends that of the sum of the first k cubes.

Original entry on oeis.org

0, 1, 5, 25, 76, 376, 500, 625, 876, 1876, 2500, 5001, 5625, 9376, 15625, 25001, 40625, 50001, 62500, 65625, 71876, 75001, 90625, 109376, 171876, 265625, 375001, 390625, 500001, 765625, 875001, 890625, 1171876, 2265625, 2890625, 4062500, 4375001, 5000001

Offset: 1

Robert Dawson, Mar 28 2018

For j >= 3, 1 + 5*10^j = A199685(j) is in the sequence, so the sequence is infinite. - Vaclav Kotesovec, Mar 29 2018
From Robert Dawson, Apr 12 2018: (Start)
This sequence is the union of the following ten subsequences.
Terms in  have fewer than d digits: they are always terms of the sequence, and always appear elsewhere, as an earlier term of the same subsequence or a related subsequence. (However, the d-th terms of the subsequences are always distinct for any d > 4.) Dashes replace certain solutions to the congruences for small values of d for which certain other divisibility criteria are not met. The integers n_0(d) and n_1(d) are the even and odd zeros of n^2+3n+4 (mod 2^d) (note that by Hensel's Lemma these always exist and each is unique).
(i)   p(d) satisfying 2^d| p(d) - n_0(d), 5^d |p(d):
  (0,<0>,500,2500,62500,62500,4062500,14062500,...)
(ii)  q(d) satisfying 2^{d-1}|q(d)-1, 5^d|q(d) for d != 3:
  (0,25,-,<625>,40625,390625,2890625,12890625,...)
(iii) q(d) + 5x10^{d-1} for d != 2:
  (5,-, 625,5625,90625, 890625,7890625, 62890625,...)
(iv)  q'(d) satisfying 2^{d-1}|q'(d) - n_1(d), 5^d|q'(d), for d != 1,3:
  (-,25,-,<625>,15625,265625,2265625,47265625,...)
(v)   q'(d) + 5x10^{d-1} for d != 2:
  (5,-,625,5625,65625,765625,7265625,97265625,...)
(vi)  r(d) satisfying 2^d|r(d), 5^d|r(d)-1 for d >= 2
  (-,76,376,9376,<9376>,109376,7109376,87109376,...) = A016090(d)
(vii) r'(d) satisfying 2^d|r'(d) - n_0(d), 5^d|r'(d)-1 for d >= 2:
  (-,76,876,1876,71876,171876,1171876,<1171876>,...)
(viii)s(d) := 5x10^{d-1}+1 for d >= 4:
  (-,-,-,5001,50001,500001,5000001,50000001,...) = A199685(d-1)
(ix)  t(d) satisfying 2^{d-1}|t(d)-n_0(d), 5^d|t(d)-1:
  (1,<1>,<1>,<1>,25001,375001,4375001,34375001,...)
(x)   t(d) + 5x10^{d-1} for d >= 4:
  (-,-,-,5001,75001,875001,9375001,84375001,...)
For d > 4, the sequence A301912 has at most 10 and at least 5 terms with d digits. The maximum is first attained for d=7. The minimum is first attained for d=168. (End)

			The sum of the first five cubes is 225, which ends in 5, so 5 is in the sequence.

Vaclav Kotesovec, Table of n, a(n) for n = 1..61
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
Wikipedia, Hensel's lemma

Cf. A000537, A199685.

seq = {}; Do[If[StringTake[ToString[k^2*(k+1)^2/4], -StringLength[ToString[k]]] == ToString[k], seq = Join[seq, {k}]], {k, 0, 1000000}]; seq (* Vaclav Kotesovec, Mar 29 2018 *)

A301912_list, k, n = [], 1, 1
while len(A301912_list) < 100:
    if n % 10**(len(str(k))) == k:
        A301912_list.append(k)
    k += 1
    n += k**3 # Chai Wah Wu, Mar 30 2018

Corrected and extended by Vaclav Kotesovec, Mar 29 2018

cp's OEIS Frontend

A018248 The 10-adic integer x = ...1787109376 satisfies x^2 = x.

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A033819 Trimorphic numbers: n^3 ends with n. Also m-morphic numbers for all m > 5 such that m-1 is not divisible by 10 and m == 3 (mod 4).

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A216093 a(n) = 10^n - (5^(2^n) mod 10^n).

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A224474 (2*16^(5^n) - 1) mod 10^n: a sequence of trimorphic numbers ending in 1.

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A216092 a(n) = 2^(2*5^(n-1)) mod 10^n.

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A075693 Difference between 10-adic numbers defined in A018248 & A018247.

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A113627 a(n) is the smallest number k such that k and 2^k have the same last n digits. Here k may have fewer than n digits and can be padded with leading zeros (cf. A121319).

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A121319 a(n) is the smallest number k such that k and 2^k have the same last n digits. Here k must have at least n digits (cf. A113627).