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

Author

Clark Kimberling, Mar 03 2020

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
		

Crossrefs

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

Programs

Mathematica

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 *)