A098608 -id:A098608 - cp's OEIS frontend

A013715 a(n) = 10^(2*n+1).

10, 1000, 100000, 10000000, 1000000000, 100000000000, 10000000000000, 1000000000000000, 100000000000000000, 10000000000000000000, 1000000000000000000000, 100000000000000000000000, 10000000000000000000000000, 1000000000000000000000000000, 100000000000000000000000000000

Offset: 0

N. J. A. Sloane

Bisection of A011557 (powers of 10). - Michel Marcus, Jan 17 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (100).

Cf. A013708-A013714, A013716-A013729, A011557.

Cf. A005408, A098608, A013747.

[10^(2*n+1): n in [0..20]]; // Vincenzo Librandi, Jun 26 2011

seq(10^(2*n+1),n=0..11); # Nathaniel Johnston, Jun 25 2011

10^(2 Range[0, 15] + 1) (* Wesley Ivan Hurt, Jan 17 2016 *)

a(n) = 10^(2*n+1); \\ Michel Marcus, Jan 17 2016

Vec(10/(1-100*x) + O(x^100)) \\ Altug Alkan, Jan 17 2016

From Philippe Deléham, Nov 25 2008: (Start)
G.f.: 10/(1-100*x).
a(n) = 100*a(n-1), n>0; a(0)=10. (End)
From Elmo R. Oliveira, Aug 26 2024 (Start)
E.g.f.: 10*exp(100*x).
a(n) = 10*A098608(n) = A011557(A005408(n)) = A013747(n)/10^(n+1). (End)

A063010 Carryless binary square of n; also Moser-de Bruijn sequence written in binary.

Original entry on oeis.org

0, 1, 100, 101, 10000, 10001, 10100, 10101, 1000000, 1000001, 1000100, 1000101, 1010000, 1010001, 1010100, 1010101, 100000000, 100000001, 100000100, 100000101, 100010000, 100010001, 100010100, 100010101, 101000000, 101000001

Offset: 0

Henry Bottomley, Jul 03 2001

Numbers that are sums of distinct powers of 100. - David Wasserman, Feb 26 2008

			a(11)=1000101, since 11 in binary is 1011 and binary carryless sum of 1011000, 0, 10110 and 1011 is 1000101.

Michael De Vlieger, Table of n, a(n) for n = 0..8191
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.

Cf. Moser-de Bruijn sequence A000695, carryless decimal squares A059729, pre-carry binary squares A063009.

With[{k = 100}, Map[FromDigits[#, k] &, Tuples[{0, 1}, 5]]] (* Michael De Vlieger, Oct 29 2022 *)

a(n) = fromdigits(binary(n),100); \\ Ruud H.G. van Tol, Dec 05 2022

def A063010(n): return int(bin(int(bin(n)[2:],4))[2:]) # Chai Wah Wu, Apr 09 2025

a(n) = A062033(n)/10, i.e., with final zero removed.
a(n) = Sum_{k>=0} A030308(n,k)*A098608(k). - Philippe Deléham, Oct 15 2011
G.f.: (1/(1 - x))*Sum_{k>=0} 100^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

More terms from David Wasserman, Feb 26 2008

A165800 Powers of 50.

Original entry on oeis.org

1, 50, 2500, 125000, 6250000, 312500000, 15625000000, 781250000000, 39062500000000, 1953125000000000, 97656250000000000, 4882812500000000000, 244140625000000000000, 12207031250000000000000, 610351562500000000000000, 30517578125000000000000000, 1525878906250000000000000000

Offset: 0

Jaroslav Krizek, Sep 27 2009

Same as Pisot sequences E(1, 50), L(1, 50), P(1, 50), T(1, 50). Essentially same as Pisot sequences E(50, 2500), L(50, 2500), P(50, 2500), T(50, 2500). See A008776 for definitions of Pisot sequences.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 50-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (50).

Cf. A000079, A000351, A008776, A011557, A098608.

[50^n: n in [0..20]]; // Vincenzo Librandi, Nov 21 2010

50^Range[0, 20] (* Paolo Xausa, Jul 09 2025 *)

A165800(n):=50^n$
makelist(A165800(n),n,0,30); /* Martin Ettl, Nov 06 2012 */

a(n)=50^n \\ Charles R Greathouse IV, Jun 19 2015

powers(50,8) \\ Charles R Greathouse IV, Jun 19 2015

G.f.: 1/(1-50*x).
a(n) = 50^n; a(n) = 50*a(n-1) a(0)=1. - Vincenzo Librandi, Nov 21 2010
From G. C. Greubel, Apr 08 2016: (Start)
a(n) = 2^n * 5^(2*n) = A000079(n)*A000351(n)^2.
a(n) = 5^n * 10^n = A000351(n)*A011557(n). (End)
From Elmo R. Oliveira, Jul 08 2025: (Start)
E.g.f.: exp(50*x).
a(n) = A098608(n)/A000079(n). (End)

A368418 Numbers X such that X^2 + Y^2 = 10^(2*k) + 1, with X > Y > 0 and k is the decimal digit length of X-1.

Original entry on oeis.org

10, 76, 100, 980, 1000, 8824, 10000, 76249, 87551, 98020, 100000, 753424, 766424, 999800, 1000000, 7209049, 7241380, 8220640, 8463640, 9801980, 9879740, 9990280, 10000000, 77053825, 78173720, 80404255, 83754376, 84711551, 86600176, 90880001, 93094625, 93728480

Offset: 1

A.H.M. Smeets, Dec 24 2023

The values X and Y are used in finding A368416.

The number of terms for a given k is 2^(f-1), where f = A119704(2*k) is the number of prime factors of 10^(2*k) + 1.

			10 is a term since X = 10, Y = 1, k = 1 and 10^2 + 1^2 = 101.
76 is a term since X = 76, Y = 65, k = 2 and 76^2 + 65^2 = 10001.
980 is a term since X = 980, Y = 199, k = 3 and 980^2 + 199^2 = 1000001.

Frits Beukers, "Getallen - Een inleiding" (In Dutch), Epsilon Uitgaven, Amsterdam (2015).

A.H.M. Smeets, Table of n, a(n) for n = 1..67
T. Granlund, Factors of 10^n + 1.
Alf van der Poorten, The Hermite-Serret Algorithm and 12^2 + 33^2. In: Lam, KY., Shparlinski, I., Wang, H., Xing, C. (eds) Cryptography and Computational Number Theory. Progress in Computer Science and Applied Logic, vol 20. Birkhäuser, Basel.

Cf. A098608, A119704, A368416.

A098609 a(n) = 100^n - 1.

Original entry on oeis.org

0, 99, 9999, 999999, 99999999, 9999999999, 999999999999, 99999999999999, 9999999999999999, 999999999999999999, 99999999999999999999, 9999999999999999999999, 999999999999999999999999, 99999999999999999999999999, 9999999999999999999999999999, 999999999999999999999999999999

Offset: 0

Henry Bottomley, Sep 17 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (101,-100).

Cf. A002283, A062397, A098608.

[100^n-1: n in [0..20]]; // Vincenzo Librandi, Sep 23 2016

Table[100^n - 1, {n, 0, 20}] (* Vincenzo Librandi, Sep 23 2016 *)

a(n) = 100*a(n-1) + 99 = A002283(2*n) = A098608(n) - 1.
From Chai Wah Wu, Sep 22 2016: (Start)
a(n) = 101*a(n-1) - 100*a(n-2) for n > 1.
G.f.: 99*x/((x - 1)*(100*x - 1)). (End)
E.g.f.: exp(x)*(exp(99*x) - 1). - Stefano Spezia, Aug 05 2024
a(n) = A002283(n)*A062397(n). - Elmo R. Oliveira, Sep 09 2024

A346460 Square array read by downward antidiagonals in which row n lists all numbers k for which all positive integers cannot be colored with two colors without any positive integer x being the same color as nx or kx (for n >= 2).

Original entry on oeis.org

4, 16, 9, 64, 81, 2, 256, 729, 8, 25, 1024, 6561, 16, 625, 36, 4096, 59049, 32, 15625, 1296, 49, 16384, 531441, 128, 390625, 46656, 2401, 4, 65536, 4782969, 256, 9765625, 1679616, 117649, 16, 3, 262144, 43046721, 512, 244140625, 60466176, 5764801, 64, 27, 100

Offset: 2

M. Eren Kesim, Aug 25 2021

Row n lists all positive integers k for which there exists at least one pair of positive integers (x, y) such that n^x = k^y and x+y is odd.
If n is an element of A007916, then row n lists all perfect powers of n^2.
A positive integer k is in row n if and only if there exists a positive integer x for which A052410(n)^x = k and A007814(A052409(n)) != A007814(x).

			Table begins:
     4,    16,      64,       256,        1024,          4096,           16384, ...
     9,    81,     729,      6561,       59049,        531441,         4782969, ...
     2,     8,      16,        32,         128,           256,             512, ...
    25,   625,   15625,    390625,     9765625,     244140625,      6103515625, ...
    36,  1296,   46656,   1679616,    60466176,    2176782336,      7836416409, ...
    49,  2401,  117649,   5764801,   282475249,   13841287201,    678223072849, ...
     4,    16,      64,       256,        1024,          4096,           16384, ...
     3,    27,      81,       243,        2187,          6561,           19683, ...
   100, 10000, 1000000, 100000000, 10000000000, 1000000000000, 100000000000000, ...

M. Eren Kesim, First Python program for A346460
M. Eren Kesim, Second Python program for A346460

First few rows are A000302, A001019, A346461, A009969, A009980, and A087752.

Cf. A007814, A007916, A013708-A013715, A052409, A052410, A098608, A346459.

Python
```
# See links.
```

A013792 a(n) = 10^(4*n + 1).

Original entry on oeis.org

10, 100000, 1000000000, 10000000000000, 100000000000000000, 1000000000000000000000, 10000000000000000000000000, 100000000000000000000000000000, 1000000000000000000000000000000000, 10000000000000000000000000000000000000, 100000000000000000000000000000000000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (10000).

Subsequence of A011557.

Cf. A013715, A013776, A013782, A016813, A098608.

[10^(4*n+1): n in [0..10]]; // Vincenzo Librandi, Jun 28 2011

a(n) = 10000*a(n-1). - Wesley Ivan Hurt, Apr 19 2023
From Elmo R. Oliveira, Aug 30 2024: (Start)
G.f.: 10/(1 - 10000*x).
E.g.f.: 10*exp(10000*x).
a(n) = A011557(A016813(n)) = A098608(n)*A013715(n) = A013776(n)*A013782(n). (End)

A013793 a(n) = 10^(4*n + 3).

Original entry on oeis.org

1000, 10000000, 100000000000, 1000000000000000, 10000000000000000000, 100000000000000000000000, 1000000000000000000000000000, 10000000000000000000000000000000, 100000000000000000000000000000000000, 1000000000000000000000000000000000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (10000).

Cf. A004767, A011557, A013715, A013777, A013783, A098608.

[10^(4*n+3): n in [0..10]]; // Vincenzo Librandi, Jun 28 2011

10^(4*Range[0, 10] + 3) (* or *)
NestList[10000*# &, 1000, 10] (* Paolo Xausa, Jul 21 2025 *)

makelist(10^(4*n+3),n,0,20); /* Martin Ettl, Oct 21 2012 */

From Elmo R. Oliveira, Aug 30 2024: (Start)
G.f.: 1000/(1 - 10000*x).
E.g.f.: 1000*exp(10000*x).
a(n) = 10000*a(n-1) for n > 0.
a(n) = A011557(A004767(n)) = A013715(n)*A098608(n+1) = A013777(n)*A013783(n). (End)

A133851 Sloping binary representation of powers of 4 (A000302), slope = -1 .

Original entry on oeis.org

1, 0, 0, 4, 0, 0, 16, 0, 0, 64, 0, 0, 256, 0, 0, 1024, 0, 0, 4096, 0, 0, 16384, 0, 0, 65536, 0, 0, 262144, 0, 0, 1048576, 0, 0, 4194304, 0, 0, 16777216, 0, 0, 67108864, 0, 0, 268435456, 0, 0, 1073741824, 0, 0, 4294967296, 0, 0, 17179869184, 0, 0

Offset: 0

Philippe Deléham, Jan 06 2008

			When powers of 4 are written in binary (see A098608), under each other as:
0000000000001 (1)
0000000000100 (4)
0000000010000 (16)
0000001000000 (64)
0000100000000 (256)
0010000000000 (1024)
1000000000000 (4096)
and one collects their bits from the column=0 to NW-direction (from the least to the most significant end), one gets 1 (1), 00 (0), 000 (0), 0100 (4), 00000 (0), 000000 (0), 0010000 (16), etc. (see 0105033 for similar transformation done on nonnegative integers)

Cf. A037095, A077957, A105033, A000302, A098608, A102370(sloping binary numbers).

a(3n) = A000302(n), a(3n+1) = a(3n+2) = 0. - Alois P. Heinz, Dec 10 2020

A177019 a(n) = 310^(2n) + 3*10^n + 1.

Original entry on oeis.org

7, 331, 30301, 3003001, 300030001, 30000300001, 3000003000001, 300000030000001, 30000000300000001, 3000000003000000001, 300000000030000000001, 30000000000300000000001, 3000000000003000000000001, 300000000000030000000000001, 30000000000000300000000000001, 3000000000000003000000000000001

Offset: 0

Vincenzo Librandi, May 24 2010

			For n=0, a(0)=7; n=1, a(1)=3*10^2+3*10+1=331.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A011557, A098608.

Magma
```
[(3*10^(2*n)+3*10^n+1): n in [0..15]];
```

CoefficientList[Series[(7 - 446 x + 1330 x^2)/((1 - x)(1 - 10 x) (1 - 100 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 19 2014 *)

G.f.: (7-446*x+1330*x^2)/((1-x)*(1-10*x)*(1-100*x)). - Vincenzo Librandi, Aug 19 2014

E.g.f.: exp(x)*(1 + 3*exp(9*x) + 3*exp(99*x)). - Stefano Spezia, Aug 05 2024

cp's OEIS Frontend

A013715 a(n) = 10^(2*n+1).

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A063010 Carryless binary square of n; also Moser-de Bruijn sequence written in binary.

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A165800 Powers of 50.

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A368418 Numbers X such that X^2 + Y^2 = 10^(2*k) + 1, with X > Y > 0 and k is the decimal digit length of X-1.

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A098609 a(n) = 100^n - 1.

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A346460 Square array read by downward antidiagonals in which row n lists all numbers k for which all positive integers cannot be colored with two colors without any positive integer x being the same color as n*x or k*x (for n >= 2).

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A013792 a(n) = 10^(4*n + 1).

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A346460 Square array read by downward antidiagonals in which row n lists all numbers k for which all positive integers cannot be colored with two colors without any positive integer x being the same color as nx or kx (for n >= 2).

A177019 a(n) = 310^(2n) + 3*10^n + 1.