A332937 -id:A332937 - cp's OEIS frontend

A332938 Indices of the primitive rows of the Wythoff array (A035513); see Comments.

1, 2, 6, 7, 8, 10, 11, 12, 14, 17, 18, 20, 21, 23, 24, 26, 27, 30, 32, 33, 36, 37, 38, 39, 40, 42, 44, 46, 48, 49, 50, 53, 54, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 79, 80, 81, 84, 85, 86, 88, 90, 92, 94, 95, 98, 100, 101, 102, 104, 107

Offset: 1

Clark Kimberling, Mar 03 2020

In a row of the Wythoff array, either every two consecutive terms are relatively prime or else no two consecutive terms are relatively prime. In the first case, we call the row primitive; otherwise, the row is an integer multiple of a tail of a preceding row. Conjectures: the maximal number of consecutive primitive rows is 5, and the limiting proportion of primitive rows exists and is approximately 0.608.

			The Wythoff array begins:
   1    2    3    5    8   13   21   34   55   89  144 ...
   4    7   11   18   29   47   76  123  199  322  521 ...
   6   10   16   26   42   68  110  178  288  466  754 ...
   9   15   24   39   63  102  165  267  432  699 1131 ...
  12   20   32   52   84  136  220  356  576  932 1508 ...
  14   23   37   60   97  157  254  411  665 1076 1741 ...
  17   28   45   73  118  191  309  500  809 1309 2118 ...
  19   31   50   81  131  212  343  555  898 1453 2351 ...
  22   36   58   94  152  246  398  644 1042 1686 2728 ...
Row 1: A000045 (Fibonacci numbers, a primitive row)
Row 2: A000032 (Lucas numbers, primitive)
Row 3: 2 times a tail of row 1
Row 4: 3 times a tail of row 1
Row 5  4 times a tail of row 1
Row 6:  essentially A000285, primitive
Row 7:  essentially A022095, primitive
Row 8:  essentially A013655, primitive
Row 9:  2 times a tail of row 2
Thus first five terms of (a(n)) are 1,2,6,7,8.

Cf. A000045, A173027, A173028, A035513, A332937.

W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k]; (* A035513 *)
t = Table[GCD[W[n, 1], W[n, 2]], {n, 1, 160}]  (* A332937 *)
Flatten[Position[t, 1]]  (* A332938 *)

A333028 Array consisting of the primitive rows of the Wythoff array (A035513), read by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 7, 14, 5, 11, 23, 17, 8, 18, 37, 28, 19, 13, 29, 60, 45, 31, 25, 21, 47, 97, 73, 50, 41, 27, 34, 76, 157, 118, 81, 66, 44, 30, 55, 123, 254, 191, 131, 107, 71, 49, 35, 89, 199, 411, 309, 212, 173, 115, 79, 57, 43, 144, 322, 665, 500, 343, 280

Offset: 1

Clark Kimberling, Mar 10 2020

In a row of the Wythoff array, either every two consecutive terms are relatively prime or else no two consecutive terms are relatively prime. In the first case, we call the row primitive; otherwise, the row is an integer multiple of a tail of a preceding row. The primitive rows are interspersed, in the sense that if h < k then the numbers in row k are interspersed, in magnitude, among numbers in row h. In each row, every pair of consecutive numbers is a Wythoff pair of relatively prime numbers. The array includes every prime.

			Northwest corner:
   1   2   3    5    8   13  21    34
   4   7  11   18   29   47  76   123
  14  23  37   60   97  157  254  411
  17  28  45   73  118  191  309  500
  19  31  50   81  131  212  343  555
  25  41  66  107  173  280  453  733
  27  44  71  115  186  301  487  788
  30  49  79  128  207  335  542  877

Cf. A000045, A000032, A332937, A332938, A333029, A333086.

W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
t = Table[GCD[W[n, 1], W[n, 2]], {n, 1, 160}]
u = Flatten[Position[t, 1]]; v[n_, k_] := W[u[[n]], k];
TableForm[Table[v[n, k], {n, 1, 30}, {k, 1, 8}]] (* A333028 array *)
Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (* A333028 sequence *)

A360374 Indices of the nonprimitive rows of the Wythoff array (A035513); see Comments.

Original entry on oeis.org

3, 4, 5, 9, 13, 15, 16, 19, 22, 25, 28, 29, 31, 34, 35, 41, 43, 45, 47, 51, 52, 55, 56, 57, 58, 61, 67, 71, 73, 77, 78, 82, 83, 87, 89, 91, 93, 96, 97, 99, 103, 105, 106, 109, 113, 115, 119, 121, 125, 129, 130, 135, 136, 137, 139, 141, 145, 151, 153, 154

Offset: 1

Clark Kimberling, Feb 04 2023

In a row of the Wythoff array, either every two consecutive terms are relatively prime or else no two consecutive terms are relatively prime. In the first case, we call the row primitive, as in A332938. Otherwise, the row is an integer multiple of a tail of a preceding row, and we call the row nonprimitive. Conjecture: the limiting proportion of nonprimitive rows exists and is approximately 0.392.

			(See A332938.)

Cf. A000045, A035513, A332937, A332938.

W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
Select[Range[200], GCD[W[#, 1], W[#, 2]] > 1 &]

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A332938 Indices of the primitive rows of the Wythoff array (A035513); see Comments.

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A333028 Array consisting of the primitive rows of the Wythoff array (A035513), read by antidiagonals.

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A360374 Indices of the nonprimitive rows of the Wythoff array (A035513); see Comments.

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