cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-14 of 14 results.

A197309 Divisors of the 9th repunit 111111111.

Original entry on oeis.org

1, 3, 9, 37, 111, 333, 333667, 1001001, 3003003, 12345679, 37037037, 111111111
Offset: 1

Views

Author

M. F. Hasler, Oct 13 2011

Keywords

Comments

The prime factorization of (10^9 - 1)/9 is 3^2 * 37 * 333667. - Alonso del Arte, Apr 27 2014

Links

Crossrefs

Cf. A109492, A197308, A070529, A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895.

Programs

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

Original entry on oeis.org

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

Views

Author

Martin Renner, Jun 08 2013

Keywords

Comments

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).

Examples

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

Links

Crossrefs

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.

Programs

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

A135365 a(n) = 3*a(n-1) - a(n-3) + 3*a(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

Views

Author

Paul Curtz, Dec 09 2007

Keywords

Links

Crossrefs

Cf. A018282, A027894, A106305.

Programs

  • Mathematica
    Join[{1, 3}, LinearRecurrence[{3, 0, -1, 3}, {9, 11, 33, 99}, 25]] (* G. C. Greubel, Oct 11 2016 *)
  • PARI
    Vec((1-15*x^3)/((1+x)*(1-3*x)*(1-x+x^2)) + O(x^40)) \\ Colin Barker, Feb 10 2016

Formula

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

A197318 Divisors of the repunit 111111111111 = A002275(12).

Original entry on oeis.org

1, 3, 7, 11, 13, 21, 33, 37, 39, 77, 91, 101, 111, 143, 231, 259, 273, 303, 407, 429, 481, 707, 777, 1001, 1111, 1221, 1313, 1443, 2121, 2849, 3003, 3333, 3367, 3737, 3939, 5291, 7777, 8547, 9191, 9901, 10101, 11211, 14443, 15873, 23331, 26159, 27573, 29703
Offset: 1

Views

Author

M. F. Hasler, Oct 13 2011

Keywords

Comments

The sequence is marked "full" since even though they don't fit into the three lines above, all 128 terms are known and available in the b-file or using the given PARI code.

Links

Crossrefs

Cf. A109492, A197308, A197309, A070529, A018282, A018766, A027890-A027895, A027889.

Programs

  • Mathematica
    Divisors[111111111111] (* Paolo Xausa, Jul 04 2024 *)
  • PARI
    divisors(1e12\9)
Previous Showing 11-14 of 14 results.

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