A272066 -id:A272066 - cp's OEIS frontend

A059988 a(n) = (10^n - 1)^2.

0, 81, 9801, 998001, 99980001, 9999800001, 999998000001, 99999980000001, 9999999800000001, 999999998000000001, 99999999980000000001, 9999999999800000000001, 999999999998000000000001, 99999999999980000000000001, 9999999999999800000000000001, 999999999999998000000000000001

Offset: 0

Henry Bottomley, Mar 07 2001

From James D. Klein, Feb 05 2012: (Start)
The periods of the reciprocals of a(n) are the consecutive integers from 0 to 10^n-1, omitting the one integer 10^n-2, right-justified in field widths of size n.
E.g.:
  1/81 = 0.012345679...
  1/9801 = 0.000102030405060708091011...9799000102...
  1/998001 = 0.000001002003004005...997999000001002... (End)
Sum of first 10^n - 1 odd numbers. - Arkadiusz Wesolowski, Jun 12 2013

			From _Reinhard Zumkeller_, May 31 2010: (Start)
n=1: ..................... 81 = 9^2;
n=2: ................... 9801 = 99^2;
n=3: ................. 998001 = 999^2;
n=4: ............... 99980001 = 9999^2;
n=5: ............. 9999800001 = 99999^2;
n=6: ........... 999998000001 = 999999^2;
n=7: ......... 99999980000001 = 9999999^2;
n=8: ....... 9999999800000001 = 99999999^2;
n=9: ..... 999999998000000001 = 999999999^2. (End)

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 32 at p. 61.
Walther Lietzmann, Lustiges und Merkwuerdiges von Zahlen und Formen, (F. Hirt, Breslau 1921-43), p. 149.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 34.

Harry J. Smith, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A075411, A075412, A075413, A075414, A075415, A075416, A075417, A178630, A178631, A178632, A178633, A178634, A178635, A272066, A272067, A272068.
Subsequence of A238237.
Cf. A002275, A002283, A002477.

A059988:=n->(10^n-1)^2; seq(A059988(n), n=0..20); # Wesley Ivan Hurt, Nov 21 2013

Table[(10^n-1)^2, {n,0,20}] (* Wesley Ivan Hurt, Nov 21 2013 *)

a(n) = { (10^n - 1)^2 } \\ Harry J. Smith, Jul 01 2009

def a(n): return (10**n - 1)**2  # Martin Gergov, Sep 10 2022

a(n) = 81*A002477(n) = A002283(n)^2 = (9*A002275(n))^2.
a(n) = {999... (n times)}^2 = {999... (n times), 000... (n times)} - {999... (n times)}. For example, 999^2 = 999000 - 999 = 998001. - Kyle D. Balliet, Mar 07 2009
a(n) = (A002283(n-1)*10 + 8) * 10^(n-1) + 1, for n>0. - Reinhard Zumkeller, May 31 2010
From Ilya Gutkovskiy, Apr 19 2016: (Start)
O.g.f.: 81*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)).
E.g.f.: (1 - 2*exp(9*x) + exp(99*x))*exp(x). (End)
Sum_{n>=1} 1/a(n) = (log(10)*(QPolyGamma(0, 1, 1/10) - log(10/9)) + QPolyGamma(1, 1, 1/10))/log(10)^2 = 0.012448721523422795191... . - Stefano Spezia, Jul 31 2024
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3). - Elmo R. Oliveira, Aug 02 2025

A272067 a(n) = (10^n-1)^4.

Original entry on oeis.org

0, 6561, 96059601, 996005996001, 9996000599960001, 99996000059999600001, 999996000005999996000001, 9999996000000599999960000001, 99999996000000059999999600000001, 999999996000000005999999996000000001, 9999999996000000000599999999960000000001, 99999999996000000000059999999999600000000001

Offset: 0

Seiichi Manyama, Apr 19 2016

The sum of the digits of a(n) is divisible by 18. For example, 9^4 = 6561 and 6 + 5 + 6 + 1 = 18 * 1.

Number of 9 in a(n) is 2*n-2 for n > 0. - Seiichi Manyama, Sep 18 2018

			From _Seiichi Manyama_, Sep 18 2018: (Start)
n| a(n) can be divided into 4 parts for n > 1.
-+--------------------------------------------
1|        65        61
2|   9   605   9   601
3|  99  6005  99  6001
4| 999 60005 999 60001
(End)

Index entries for linear recurrences with constant coefficients, signature (11111,-11222110,1122211000,-11111000000,10000000000).

Cf. A002283, A059988, A272066, A272068, A319358.

[(10^n-1)^4 : n in [0..10]]; // Wesley Ivan Hurt, Apr 19 2016

A272067:=n->(10^n-1)^4: seq(A272067(n), n=0..15); # Wesley Ivan Hurt, Apr 19 2016

(10^Range[0, 10] - 1)^4 (* Wesley Ivan Hurt, Apr 19 2016 *)

a(n) = (10^n-1)^4; \\ Michel Marcus, Apr 19 2016

Ruby
```
(0..n).each{|i| p ('9' * i).to_i ** 4}
```

a(n) = A059988(n)^2 = A002283(n)^4.
From Ilya Gutkovskiy, Apr 19 2016: (Start)
O.g.f.: 6561*x*(1 + 100*x)*(1 + 3430*x + 10000*x^2)/((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)*(1 - 10000*x)).
E.g.f.: (1 - 4*exp(9*x) + 6*exp(99*x) - 4*exp(999*x) + exp(9999*x))*exp(x). (End)

A272068 a(n) = (10^n-1)^5.

Original entry on oeis.org

0, 59049, 9509900499, 995009990004999, 99950009999000049999, 9999500009999900000499999, 999995000009999990000004999999, 99999950000009999999000000049999999, 9999999500000009999999900000000499999999, 999999995000000009999999990000000004999999999, 99999999950000000009999999999000000000049999999999

Offset: 0

Seiichi Manyama, Apr 19 2016

The sum of the digits of a(n) is divisible by 27. For example, 9^5 = 59049 and 5 + 9 + 0 + 4 + 9 = 27 * 1.

Number of 9 in a(n) is 3*n-1 for n > 0. - Seiichi Manyama, Sep 18 2018

			From _Seiichi Manyama_, Sep 18 2018: (Start)
n| a(n) can be divided into 5 parts for n > 1.
-+--------------------------------------------
1|        5    9    04    9
2|   9   50   99   004   99
3|  99  500  999  0004  999
4| 999 5000 9999 00004 9999
(End)

Index entries for linear recurrences with constant coefficients, signature (111111,-1122322110,1123333211000,-112232211000000,1111110000000000,-1000000000000000).

Cf. A002283, A059988, A272066, A272067, A319358.

[(10^n-1)^5 : n in [0..10]]; // Wesley Ivan Hurt, Apr 19 2016

A272068:=n->(10^n-1)^5: seq(A272068(n), n=0..10); # Wesley Ivan Hurt, Apr 19 2016

(10^Range[0, 10] - 1)^5 (* Wesley Ivan Hurt, Apr 19 2016 *)

a(n) = (10^n-1)^5; \\ Michel Marcus, Apr 19 2016

Ruby
```
(0..n).each{|i| p ('9' * i).to_i ** 5}
```

a(n) = A002283(n)^5.
From Ilya Gutkovskiy, Apr 19 2016: (Start)
O.g.f.: 59049*x*(1 + 49940*x + 78366000*x^2 + 4994000000*x^3 + 10000000000*x^4)/((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)*(1 - 10000*x)*(1 - 100000*x)).
E.g.f.: -exp(x) + 5*exp(10*x) - 10*exp(100*x) + 10*exp(1000*x) - 5*exp(10000*x) + exp(100000*x). (End)

A067074 a(n) = smallest cube m^3 such that the sum of the digits of m^3 is equal to n^3.

Original entry on oeis.org

0, 1, 8, 19683, 1693669888, 39677989979796875, 56984998629886989599887999587

Offset: 0

Amarnath Murthy, Jan 05 2002

If n = 6*k, a(n) <= A272066(n^3/18). - Seiichi Manyama, Aug 12 2017

			a(3) = 19683 as it is the smallest cube whose digit sum = 27 = 3^3.

Cf. A000578, A067072, A067075.

a(n) = A067075(n)^3. - R. J. Mathar, Aug 23 2018

Corrected by Stefan Steinerberger, Nov 09 2005, using existing corrections to A067075

a(0)=0 prepended by Seiichi Manyama, Aug 12 2017

A297785 Decimal expansion of 4099200041/9999^3.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jan 05 2018

			0.0041004300470053006100710083009701130131015101730197...

Paolo Xausa, Table of n, a(n) for n = 0..10000
John D. Cook, Prime generating fractions
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A005846, A202018, A272066, A297786, A297787.

First[RealDigits[4099200041/9999^3, 10, 100, -1]] (* Paolo Xausa, Jun 16 2024 *)

Sum_{k>=0} 10^(-4*k-4)*A202018(k) = 4099200041/9999^3.

A297786 Decimal expansion of 10980011/999^3.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jan 06 2018

			0.011013017023031041053067083101121...

John D. Cook, Prime generating fractions

Cf. A048058, A048059, A272066, A297785, A297787.

Join[{0},RealDigits[10980011/999^3,10,120][[1]]] (* Harvey P. Dale, Aug 20 2021 *)

Sum_{k>=0} 10^(-3*k-3)*A048058(k) = 10980011/999^3.

A297787 Decimal expansion of 16968017/999^3.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jan 06 2018

			0.017019023029037047059073089107127149173199227257289...

John D. Cook, Prime generating fractions

Cf. A007635, A272066, A297785, A297786.

RealDigits[16968017/999^3, 10, 111][[1]] (* or *)
RealDigits[ Sum[10^(-3k -3)*(k^2 +k +17), {k, 0, 37}], 10, 111][[1]] (* Robert G. Wilson v, Jan 07 2018 *)

Sum_{k>=0} 10^(-3*k-3)*(k^2+k+17) = 16968017/999^3.

cp's OEIS Frontend

A059988 a(n) = (10^n - 1)^2.

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A272067 a(n) = (10^n-1)^4.

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A272068 a(n) = (10^n-1)^5.

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A067074 a(n) = smallest cube m^3 such that the sum of the digits of m^3 is equal to n^3.

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A297785 Decimal expansion of 4099200041/9999^3.

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Mathematica

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A297786 Decimal expansion of 10980011/999^3.

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Mathematica

Formula

A297787 Decimal expansion of 16968017/999^3.

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