A027894 -id:A027894 - cp's OEIS frontend

A226477 Table (read by rows) of the natural numbers (in ascending order) whose reciprocals have only periodic decimals of length k.

1, 3, 9, 11, 33, 99, 27, 37, 111, 333, 999, 101, 303, 909, 1111, 3333, 9999, 41, 123, 271, 369, 813, 2439, 11111, 33333, 99999, 7, 13, 21, 39, 63, 77, 91, 117, 143, 189, 231, 259, 273, 297, 351, 407, 429, 481, 693, 777, 819, 1001, 1221, 1287, 1443, 2079, 2331, 2457, 2849, 3003, 3367, 3663, 3861, 4329, 5291, 6993, 8547, 9009, 10101, 10989, 12987, 15873, 25641, 27027, 30303, 37037, 47619, 76923, 90909, 111111, 142857, 333333, 999999

Offset: 1

Martin Renner, Jun 08 2013

The k-th row always ends with 10^k - 1 = 99..99 (k times 9).

The number of elements in row k is A059892(k).

			The table T(k,m), m = 1..A059892(k), begins
  1, 3, 9;
  11, 33, 99;
  27, 37, 111, 333, 999;
  etc.

Jianing Song, Rows n = 1..32, flattened

Cf. A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895, A027896, A027897, A109933, A106305, A111117, A111211, A113116, A113522 (Divisors of 10^k - 1, k = 2..18), A059892, A084680.

a:=[1,3,9]: S:={1,3,9}: for k from 2 to 6 do T:=numtheory[divisors](10^k-1): a:=[op(a),op(T minus S)]: S:=S union T; od: a;

Row(n) = my(v=divisors(10^n-1)); select(x->(znorder(Mod(10,x))==n), v) \\ Jianing Song, Jun 15 2021

A135365 a(n) = 3a(n-1) - a(n-3) + 3a(n-4), with initial values 1,3,9,11.

Original entry on oeis.org

1, 3, 9, 11, 33, 99, 313, 939, 2817, 8435, 25305, 75915, 227761, 683283, 2049849, 6149531, 18448593, 55345779, 166037353, 498112059, 1494336177, 4483008515, 13449025545, 40347076635, 121041229921, 363123689763, 1089371069289, 3268113207851, 9804339623553

Offset: 0

Paul Curtz, Dec 09 2007

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-1,3).

Cf. A018282, A027894, A106305.

Join[{1, 3}, LinearRecurrence[{3, 0, -1, 3}, {9, 11, 33, 99}, 25]] (* G. C. Greubel, Oct 11 2016 *)

Vec((1-15*x^3)/((1+x)*(1-3*x)*(1-x+x^2)) + O(x^40)) \\ Colin Barker, Feb 10 2016

From Richard Choulet, Jan 02 2008: (Start)
a(n) = (1/7)*3^(n+1) + (4/3)*(-1)^n - (16/21)*cos(Pi*n/3) + (16*sqrt(3)/7)*sin(Pi*n/3).
a(n) = (1/7)*3^(n+1) + (1/7)*[4; 12; 36; -4; -12; -36] for n>=0. (End)
G.f.: (1 - 15*x^3) / ((1+x)*(1-3*x)*(1-x+x^2)). - Colin Barker, Feb 10 2016

A069858 1/n has period 4 in base 10.

Original entry on oeis.org

101, 202, 303, 404, 505, 606, 808, 909, 1010, 1111, 1212, 1515, 1616, 1818, 2020, 2222, 2424, 2525, 3030, 3232, 3333, 3636, 4040, 4444, 4545, 4848, 5050, 5555, 6060, 6464, 6666, 7272, 7575, 8080, 8888, 9090, 9696, 9999, 10100, 11110, 12120, 12625

Offset: 1

Henry Bottomley, May 02 2002

			1212 is in the sequence since 1/1212=0.0008250825082508250825082508250....

Cf. A027894, A003592, A069105, A070021, A070022, A070023.

Select[Range[13000],Length[RealDigits[1/#][[1,-1]]]==4&] (* Harvey P. Dale, Jun 30 2017 *)

Numbers of the form a*b*c where a is in {101, 303, 909, 1111, 3333, 9999}, b is a power of 2 and c is a power of 5.

cp's OEIS Frontend

A226477 Table (read by rows) of the natural numbers (in ascending order) whose reciprocals have only periodic decimals of length k.

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A135365 a(n) = 3a(n-1) - a(n-3) + 3a(n-4), with initial values 1,3,9,11.

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A069858 1/n has period 4 in base 10.

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Author

Keywords

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cp's OEIS Frontend

A226477 Table (read by rows) of the natural numbers (in ascending order) whose reciprocals have only periodic decimals of length k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

A135365 a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4), with initial values 1,3,9,11.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A069858 1/n has period 4 in base 10.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A135365 a(n) = 3a(n-1) - a(n-3) + 3a(n-4), with initial values 1,3,9,11.