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A332938
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Indices of the primitive rows of the Wythoff array (A035513); see Comments.
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8
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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