A091449 -id:A091449 - cp's OEIS frontend

A091450 Transpose of A091449: array T(n,k) read by antidiagonals, where column k is the increasing sequence of numbers m for which the simple continued fraction of sqrt(m) has period k, n >= 1, k >= 0.

1, 4, 2, 9, 5, 3, 16, 10, 6, 41, 25, 17, 8, 130, 7, 36, 26, 11, 269, 14, 13, 49, 37, 12, 370, 23, 29, 19, 64, 50, 15, 458, 28, 53, 21, 58, 81, 65, 18, 697, 32, 74, 22, 73, 31, 100, 82, 20, 986, 33, 85, 45, 202, 44, 106, 121, 101, 24, 1313, 34, 89, 52, 250, 69, 113, 43

Offset: 1

Clark Kimberling, Feb 03 2004

A permutation of the positive integers.

			Array begins:
  n\k|   0   1  2    3  4   5  6    7   8   9  10
  ---+-------------------------------------------
   1 |   1   2  3   41  7  13 19   58  31 106  43
   2 |   4   5  6  130 14  29 21   73  44 113  67
   3 |   9  10  8  269 23  53 22  202  69 137  86
   4 |  16  17 11  370 28  74 45  250  71 149  93
   5 |  25  26 12  458 32  85 52  274  91 265 115
   6 |  36  37 15  697 33  89 54  314  92 389 116
   7 |  49  50 18  986 34 125 57  349 108 493 118
   8 |  64  65 20 1313 47 173 59  425 135 610 129
   9 |  81  82 24 1325 55 185 70  538 153 698 154
  10 | 100 101 27 1613 60 218 77  761 158 754 159
  11 | 121 122 30 1714 62 229 88 1010 160 970 161
The least m for which CF(sqrt(m)) has period of length 4 is m=7, with CF=[2;1,1,1,4,1,1,1,4,1,1,1,4,...]; thus T(1,4)=7.
[The array T(n,k) is indexed by n=1,2,3,..., k=0,1,2,3... .]
Column 0 consists of squares: 1,4,9,...

Cf. A002522, A013642, A091449, A091451, A091453.

a(19) = T(3,3) corrected by Pontus von Brömssen, Nov 23 2024

A091453 Triangular array T(n,k) read by rows in which row n consists of the numbers floor(2n/k), k=1,2,...,2n+1.

Original entry on oeis.org

0, 0, 2, 1, 0, 4, 2, 1, 1, 0, 6, 3, 2, 1, 1, 1, 0, 8, 4, 2, 2, 1, 1, 1, 1, 0, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 0, 12, 6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 0, 14, 7, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 18, 9, 6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 20, 10, 6

Offset: 0

Clark Kimberling, Feb 03 2004

			First five rows:
  0
  0 2 1
  0 4 2 1 1
  0 6 3 2 1 1 1
  0 8 4 2 2 1 1 1 1

Andrew Howroyd, Table of n, a(n) for n = 0..10201

Cf. A013942 (without first column).

Cf. A091449, A327898.

T(n, k) = 2*n\k;
tabf(nn) = for (n=0, nn, for (k=1, 2*n+1, print1(T(n,k), ", ")); print()); \\ Michel Marcus, Sep 30 2016

a(n)={if(n<1, 0, my(t=sqrtint(n-1)); 2*t\(n-t^2))} \\ Andrew Howroyd, Oct 02 2019

a(n) = floor(1/(sqrt(n)-floor(sqrt(n)))) for n not a square; a(n) = 0 otherwise. - Andrew Howroyd, Oct 02 2019

Offset corrected and missing a(99) inserted by Andrew Howroyd, Oct 02 2019

A091451 Array T(n,k) read by antidiagonals: (row 0)=squares, (row 1)={numbers m for which the simple continued fraction (CF) of sqrt(m) has period length 1}; once (row n) is defined, let (row n+1) begin with the least positive integer not already in a row and let the rest of (row n+1) be the other m's for which CF(sqrt(m)) has the same period length.

Original entry on oeis.org

1, 2, 4, 3, 5, 9, 7, 6, 10, 16, 13, 14, 8, 17, 25, 19, 29, 23, 11, 26, 36, 31, 21, 53, 28, 12, 37, 49, 41, 44, 22, 74, 32, 15, 50, 64, 43, 130, 69, 45, 85, 33, 18, 65, 81, 46, 67, 269, 71, 52, 89, 34, 20, 82, 100, 58, 76, 86, 370, 91, 54, 125, 47, 24, 101, 121

Offset: 0

Clark Kimberling, Feb 03 2004

A permutation of the positive integers.
From Pontus von Brömssen, Nov 23 2024: (Start)
Rows of A091449 sorted by the first term.
First column gives indices of new terms of A003285.
(End)

			7 is the least positive integer not in rows 0,1,2, so 7=T(3,0); the period length of sqrt(7) is 4, as is the case with T(3,1)=14, T(3,2)=23, etc.
Corner:
  1    4    9    16    25    36   49   64
  2    5   10    17    26    37   50   65
  3    6    8    11    12    15   18   20
  7   14   23    28    32    33   34   47
 13   29   53    74    85    89  125  173
 19   21   22    45    52    54   57   59

Cf. A003285, A091449, A091453.

Map[Map[#[[1]] &, #] &,
 GatherBy[Map[{#, Flatten[ContinuedFraction[Sqrt[#]]]} &, Range[500]],
   Length[#[[2]]] &]]  (* Peter J. C. Moses, May 11 2023 *)

a(47) = T(7,2) corrected by Clark Kimberling, May 20 2023

More terms from Pontus von Brömssen, Nov 23 2024

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A091450 Transpose of A091449: array T(n,k) read by antidiagonals, where column k is the increasing sequence of numbers m for which the simple continued fraction of sqrt(m) has period k, n >= 1, k >= 0.

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A091453 Triangular array T(n,k) read by rows in which row n consists of the numbers floor(2n/k), k=1,2,...,2n+1.

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