A198971 -id:A198971 - cp's OEIS frontend

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).

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 *)

A099150 Positive integers k such that f(k)+f(k)=concatenation of k and k, where f(k)=k(k+3)/2 (A000096).

Original entry on oeis.org

8, 98, 998, 9998, 99998, 999998, 9999998, 99999998, 999999998, 9999999998, 99999999998, 999999999998, 9999999999998, 99999999999998, 999999999999998, 9999999999999998, 99999999999999998, 999999999999999998, 9999999999999999998, 99999999999999999998

Offset: 1

John W. Layman, Sep 30 2004

By the definition, k*(k+3) = k*10^m+k. So k+3 = 10^m+1, that is k = 10^m-2. - Seiichi Manyama, Aug 31 2019

			99998*(99998+3) = 9999899998 (concatenation of 99998 and 99998).

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A000096, A002283, A096032, A099148, A099149, A177096, A198971.

for(k=1, 1e9, if(k*(k+3)==eval(Str(k, k)), print1(k", "))) \\ Seiichi Manyama, Aug 31 2019

{a(n) = 10^n-2} \\ Seiichi Manyama, Aug 31 2019

a(n) = A002283(n) - 1 = 10^n - 2. - Seiichi Manyama, Aug 31 2019
From Chai Wah Wu, Jun 15 2020: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 2.
G.f.: x*(10*x + 8)/((x - 1)*(10*x - 1)). (End)
E.g.f.: 1 - 2*exp(x) + exp(10*x). - Stefano Spezia, May 02 2025
a(n) = 2*A198971(n-1) = A177096(n)/5. - Elmo R. Oliveira, May 02 2025

a(9)-a(20) from Seiichi Manyama, Aug 31 2019

A224473 (2*5^(2^n) - 1) mod 10^n: a sequence of trimorphic numbers ending in 9.

Original entry on oeis.org

9, 49, 249, 1249, 81249, 781249, 5781249, 25781249, 425781249, 6425781249, 36425781249, 836425781249, 9836425781249, 19836425781249, 519836425781249, 2519836425781249, 12519836425781249, 512519836425781249, 4512519836425781249, 84512519836425781249

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 A091661. The other trimorphic numbers ending in 9 are included in A002283, A198971 and A224475.

def A224473(n) : return crt(1, -1, 2^n, 5^n);

a(n) = (2 * A007185(n) - 1) mod 10^n.

A086940 a(n) = k where R(k+4) = 2.

Original entry on oeis.org

16, 196, 1996, 19996, 199996, 1999996, 19999996, 199999996, 1999999996, 19999999996, 199999999996, 1999999999996, 19999999999996, 199999999999996, 1999999999999996, 19999999999999996, 199999999999999996, 1999999999999999996, 19999999999999999996

Offset: 1

Ray Chandler, Jul 24 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A004086, A086942, A086945, A198971.

[2*(10^n-2): n in [1..20] ]; // Vincenzo Librandi, Aug 22 2011

Table[FromDigits[Join[PadRight[{1},n,9],{6}]],{n,20}] (* or *) 2 (10^Range[20] - 2) (* or *) LinearRecurrence[{11,-10},{16,196},20] (* Harvey P. Dale, Aug 11 2012 *)

a(n) = 2*(10^n - 2).
R(a(n)) = A086945(n).
a(n) = 11*a(n-1) - 10*a(n-2); a(1)=16, a(2)=196. - Harvey P. Dale, Aug 11 2012
From Elmo R. Oliveira, Apr 30 2025: (Start)
G.f.: 4*x*(5*x+4)/((x-1)*(10*x-1)).
E.g.f.: 2*(1 - 2*exp(x) + exp(10*x)).
a(n) = 4*A198971(n-1) = A086942(n)/2. (End)

A086942 Integers k such that R(k+8) = 4.

Original entry on oeis.org

32, 392, 3992, 39992, 399992, 3999992, 39999992, 399999992, 3999999992, 39999999992, 399999999992, 3999999999992, 39999999999992, 399999999999992, 3999999999999992, 39999999999999992, 399999999999999992, 3999999999999999992, 39999999999999999992

Offset: 1

Ray Chandler, Jul 24 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A004086, A086940, A086943, A099150, A198971.

[4*10^n-8: n in [1..25]]; // Vincenzo Librandi, Aug 22 2011

4*10^Range[20] - 8 (* Paolo Xausa, Sep 21 2024 *)

a(n) = 4*10^n - 8.
R(a(n)) = A086943(n).
G.f.: 8*x*(5*x+4)/((10*x-1)*(x-1)).
a(n) = 8*A198971(n-1).
From Elmo R. Oliveira, May 01 2025: (Start)
E.g.f.: 4*(1 - 2*exp(x) + exp(10*x)).
a(n) = 4*A099150(n) = 2*A086940(n).
a(n) = 11*a(n-1) - 10*a(n-2) for n > 2. (End)

A224475 (2*5^(2^n) + (10^n)/2 - 1) mod 10^n: a sequence of trimorphic numbers ending (for n > 1) in 9.

Original entry on oeis.org

4, 99, 749, 6249, 31249, 281249, 781249, 75781249, 925781249, 1425781249, 86425781249, 336425781249, 4836425781249, 69836425781249, 19836425781249, 7519836425781249, 62519836425781249, 12519836425781249, 9512519836425781249, 34512519836425781249

Offset: 1

Eric M. Schmidt, Apr 07 2013

a(n) is the unique positive integer less than 10^n such that a(n) + 2^(n-1) - 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. Converges to the 10-adic number A091661. The other trimorphic numbers ending in 9 are included in A002283, A198971, and A224473.

Table[Mod[2*5^2^n+(10^n/2)-1,10^n],{n,20}] (* Harvey P. Dale, Sep 08 2024 *)

def A224475(n) : return crt(2^(n-1)+1, -1, 2^n, 5^n)

a(n) = (A224473(n) + 10^n / 2) mod 10^n.

A323639 a(n) = 3*(10^n - 4)/9.

Original entry on oeis.org

-1, 2, 32, 332, 3332, 33332, 333332, 3333332, 33333332, 333333332, 3333333332, 33333333332, 333333333332, 3333333333332, 33333333333332, 333333333333332, 3333333333333332, 33333333333333332, 333333333333333332, 3333333333333333332, 33333333333333333332

Offset: 0

Seiichi Manyama, Aug 31 2019

			        (0+1) * (3*0-1) = -1.
        (3+1) * (3*3-1) = 32.
      (33+1) * (3*33-1) = 3332.
    (333+1) * (3*333-1) = 333332.
  (3333+1) * (3*3333-1) = 33333332.
(33333+1) * (3*33333-1) = 3333333332.
-------------------------------------
        8 * 4 = 32.
      68 * 49 = 3332.
    668 * 499 = 333332.
  6668 * 4999 = 33333332.
66668 * 49999 = 3333333332.

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002277, A073555, A086948, A093137, A198971, A246057.

Table[(10^n-4)/3,{n,0,20}] (* or *) LinearRecurrence[{11,-10},{-1,2},21] (* Harvey P. Dale, Jan 09 2021 *)

PARI
```
{a(n) = 3*(10^n-4)/9}
```

N=40; x='x+O('x^N); Vec((-1+13*x)/((1-x)*(1-10*x)))

G.f.: (-1+13*x)/((1-x)*(1-10*x)).
a(n) = 11*a(n-1) - 10*a(n-2).
a(n) = A002277(n) - 1.
a(n) = 2*A246057(n-1) for n > 0.
a(2*n) = (A002277(n)+1) * (3*A002277(n)-1).
a(2*n) = A073555(n+1) * A198971(n-1) for n > 0.
E.g.f.: exp(x)*(exp(9*x) - 4)/3. - Stefano Spezia, May 02 2025
a(n) = A086948(n)/6 for n >= 1. - Elmo R. Oliveira, May 06 2025

A181607 Numbers n with k digits such that n^2 == 1 (mod 10^k).

Original entry on oeis.org

1, 9, 49, 51, 99, 249, 251, 499, 501, 749, 751, 999, 1249, 3751, 4999, 5001, 6249, 8751, 9999, 18751, 31249, 49999, 50001, 68751, 81249, 99999, 218751, 281249, 499999, 500001, 718751, 781249, 999999, 4218751, 4999999, 5000001, 5781249, 9218751

Offset: 1

Robert G. Wilson v, Nov 01 2010

Least term of n digits: 1, 49, 249, 1249, 18751, 218751, 4218751, ..., .
If n of k digits is present then 10^k-n is present.
The union of A002283, A198971, A199685, A224473, A224474, A224475, and A224476 (except that this sequence omits 0, 4, and 6). - Eric M. Schmidt, Jan 26 2016

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

Cf. A007185, A016090, A033819.

Table[ Select[ Range[10^(k - 1), 10^k - 1], Mod[ #^2, 10^k] == 1 &], {k, 7}] // Flatten

A126687 Maximal value of GCD of 2 distinct n-digit numbers using the same digits.

Original entry on oeis.org

12, 243, 2475, 28575, 428571, 4928571, 49928571

Offset: 2

Dmitry Tishkov (DimaTishkov(AT)yandex.ru), Feb 14 2007

Without the "same digits" constraint, one would get A198971. - Michel Marcus, Jun 24 2013

			a(3)=243=GCD(729,972).

A177096 a(n) = 5*(10^n - 2).

Original entry on oeis.org

40, 490, 4990, 49990, 499990, 4999990, 49999990, 499999990, 4999999990, 49999999990, 499999999990, 4999999999990, 49999999999990, 499999999999990, 4999999999999990, 49999999999999990, 499999999999999990, 4999999999999999990, 49999999999999999990, 499999999999999999990

Offset: 1

Vincenzo Librandi

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A099150, A198971.

Vec(10*x*(4+5*x)/(10*x^2-11*x+1) + O(x^21)) \\ Elmo R. Oliveira, May 02 2025

G.f.: 10*x*(4+5*x)/((10*x-1)*(x-1)). - R. J. Mathar, Jan 06 2011
From Elmo R. Oliveira, May 02 2025: (Start)
E.g.f.: 5*(1 + exp(10*x) - 2*exp(x)).
a(n) = 5*A099150(n) = 10*A198971(n-1).
a(n) = 11*a(n-1) - 10*a(n-2) for n > 2. (End)

More terms from Elmo R. Oliveira, May 02 2025

cp's OEIS Frontend

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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A099150 Positive integers k such that f(k)+f(k)=concatenation of k and k, where f(k)=k(k+3)/2 (A000096).

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A224473 (2*5^(2^n) - 1) mod 10^n: a sequence of trimorphic numbers ending in 9.

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A086940 a(n) = k where R(k+4) = 2.

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A086942 Integers k such that R(k+8) = 4.

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A224475 (2*5^(2^n) + (10^n)/2 - 1) mod 10^n: a sequence of trimorphic numbers ending (for n > 1) in 9.

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A323639 a(n) = 3*(10^n - 4)/9.

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A181607 Numbers n with k digits such that n^2 == 1 (mod 10^k).