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

%I A332938 #7 Mar 09 2020 15:18:32
%S A332938 1,2,6,7,8,10,11,12,14,17,18,20,21,23,24,26,27,30,32,33,36,37,38,39,
%T A332938 40,42,44,46,48,49,50,53,54,59,60,62,63,64,65,66,68,69,70,72,74,75,76,
%U A332938 79,80,81,84,85,86,88,90,92,94,95,98,100,101,102,104,107
%N A332938 Indices of the primitive rows of the Wythoff array (A035513); see Comments.
%C A332938 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.
%e A332938 The Wythoff array begins:
%e A332938    1    2    3    5    8   13   21   34   55   89  144 ...
%e A332938    4    7   11   18   29   47   76  123  199  322  521 ...
%e A332938    6   10   16   26   42   68  110  178  288  466  754 ...
%e A332938    9   15   24   39   63  102  165  267  432  699 1131 ...
%e A332938   12   20   32   52   84  136  220  356  576  932 1508 ...
%e A332938   14   23   37   60   97  157  254  411  665 1076 1741 ...
%e A332938   17   28   45   73  118  191  309  500  809 1309 2118 ...
%e A332938   19   31   50   81  131  212  343  555  898 1453 2351 ...
%e A332938   22   36   58   94  152  246  398  644 1042 1686 2728 ...
%e A332938 Row 1: A000045 (Fibonacci numbers, a primitive row)
%e A332938 Row 2: A000032 (Lucas numbers, primitive)
%e A332938 Row 3: 2 times a tail of row 1
%e A332938 Row 4: 3 times a tail of row 1
%e A332938 Row 5  4 times a tail of row 1
%e A332938 Row 6:  essentially A000285, primitive
%e A332938 Row 7:  essentially A022095, primitive
%e A332938 Row 8:  essentially A013655, primitive
%e A332938 Row 9:  2 times a tail of row 2
%e A332938 Thus first five terms of (a(n)) are 1,2,6,7,8.
%t A332938 W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k]; (* A035513 *)
%t A332938 t = Table[GCD[W[n, 1], W[n, 2]], {n, 1, 160}]  (* A332937 *)
%t A332938 Flatten[Position[t, 1]]  (* A332938 *)
%Y A332938 Cf. A000045, A173027, A173028, A035513, A332937.
%K A332938 nonn,easy
%O A332938 1,2
%A A332938 _Clark Kimberling_, Mar 03 2020