A332938 -id:A332938 - cp's OEIS frontend

A332937 a(n) is the greatest common divisor of the first two terms of row n of the Wythoff array (A035513).

1, 1, 2, 3, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 5, 1, 1, 6, 1, 1, 7, 1, 1, 8, 1, 1, 9, 2, 1, 10, 1, 1, 11, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 5, 4, 3, 1, 1, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 2, 1, 1, 1, 4, 7, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 8, 1, 3, 1, 2, 1, 1, 5, 12, 1, 2, 1, 1, 1, 2, 1, 13, 3, 1, 1

Offset: 1

Clark Kimberling, Mar 03 2020

a(n) is also the gcd of every pair of consecutive terms of row n of the Wythoff array. Conjectures: the maximal number of consecutive 1's is 5, and the limiting proportion of 1's exists. See A332938.

If seems that for all primes p > 3, a(1+p) = 1. - Antti Karttunen, Jan 15 2025

			See A332938.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000045, A173027, A173028, A035513, A332938 (positions of 1's).

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 *)

T(n, k) = (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k); \\ A035513
a(n) = gcd(T(n, 0), T(n, 1)); \\ Michel Marcus, Mar 03 2020

More terms from Antti Karttunen, Jan 15 2025

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 *)

A360379 a(n) = number of the antidiagonal of the Wythoff array (A035513) that includes prime(n).

Original entry on oeis.org

2, 3, 4, 3, 4, 6, 7, 8, 7, 6, 9, 8, 11, 17, 7, 21, 23, 24, 26, 13, 10, 14, 21, 10, 10, 39, 40, 13, 27, 19, 49, 12, 53, 23, 17, 37, 11, 63, 41, 14, 69, 29, 12, 47, 76, 10, 81, 35, 55, 88, 12, 92, 18, 26, 40, 101, 65, 104, 67, 108, 44, 30, 118, 75, 120, 22

Offset: 1

Clark Kimberling, Feb 05 2023

Conjecture: Only a finite number of positive integers are missing.

			The first 6 antidiagonals of the Wythoff array are (1), (2,4), (3,7,6), (5,11,10,9), (8,18,16,15,12), (12,29,26,24,20,14). The 10th prime is 29, which occurs in antidiagonal 6, so a(10) = 6.

Cf. A000040, A035513, A332938, A360376, A360377, A360378, A360380.

W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
t = Table[W[n - k + 1, k], {n, 300}, {k, n, 1, -1}];
Map[#[[1]] &, Most[Reap[NestWhileList[# + 1 &, 1,
Length[Sow[FirstPosition[t, Prime[#]]]] > 1 &]][[2]][[1]]]]
(* Peter J. C. Moses, Feb 08 2023 *)

A360377 a(n) = number of the row of the Wythoff array (A035513) that includes prime(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 7, 8, 6, 2, 8, 6, 10, 17, 2, 21, 23, 24, 26, 11, 7, 12, 20, 1, 6, 39, 40, 10, 26, 17, 49, 8, 53, 21, 14, 36, 6, 63, 40, 10, 69, 27, 7, 46, 76, 2, 81, 33, 54, 88, 1, 92, 14, 23, 38, 64, 66, 42, 27, 74, 18, 80, 84, 53, 54, 90, 94, 59, 60, 24

Offset: 1

Clark Kimberling, Feb 04 2023

Conjecture: every primitive row number, as defined in A332938, occurs infinitely many times in this sequence.

			The 10th prime is 29, which occurs in row 7, so a(10) = 2.

Cf. A000040, A035513, A332938, A360378, A360379, A360380.

W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
t = Table[W[n, k], {n, 100}, {k, 1, 20}];
a[n_] := Select[Range[100], MemberQ[t[[#]], Prime[n]] &]
Flatten[Table[a[n], {n, 1, 100}]]

Every prime p has a unique representation p = p(m,k) = F(k+1)*[m*tau] + (m-1)*F(k), where F(h) = A000045(h) = h-th Fibonacci number, [ ] = floor, and tau = (1+sqrt(5))/2 = golden ratio, as in A001622. Here, a(n) is the number m such that prime(n) = p(m,k) for some k.

A360378 a(n) = number of the column of the Wythoff array (A035513) that includes prime(n).

Original entry on oeis.org

2, 3, 4, 2, 3, 6, 1, 1, 2, 5, 2, 3, 2, 1, 6, 1, 1, 1, 1, 3, 4, 3, 2, 10, 5, 1, 1, 4, 2, 3, 1, 5, 1, 3, 4, 2, 6, 1, 2, 5, 1, 3, 6, 2, 1, 9, 1, 3, 2, 1, 12, 1, 5, 4, 3, 1, 2, 1, 2, 1, 3, 4, 1, 2, 1, 5, 1, 2, 1, 1, 2, 3, 3, 1, 2, 1, 1, 2, 3, 3, 5, 2, 2, 1, 2, 3

Offset: 1

Clark Kimberling, Feb 04 2023

Conjecture: every positive integer occurs infinitely many times in this sequence.

			The 10th prime is 29, which occurs in column 5, so a(10) = 5.

Cf. A000040, A035513, A332938, A360377, A360379, A360380.

W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
t = Table[W[n, k], {k, 200}, {n, 1, 600}];
a[n_] := Select[Range[200], MemberQ[t[[#]], Prime[n]] &]
Flatten[Table[a[n], {n, 1, 100}]]

Every prime p has a unique representation p = p(m,k) = F(k+1)*[m*tau] + (m-1)*F(k), where F(h) = A000045(h) = h-th Fibonacci number, [ ] = floor, and tau = (1+sqrt(5))/2 = golden ratio, as in A001622. Here, a(n) is the number k such that prime(n) = p(m,k) for some m.

A360380 a(n) = number of the diagonal of the Wythoff array, A035513, that includes prime(n). See Comments.

Original entry on oeis.org

1, 2, 3, 0, 1, 5, -6, -7, -4, 3, -6, -3, -8, -16, 4, -20, -22, -23, -25, -8, -3, -9, -18, 9, -1, -38, -39, -6, -24, -14, -48, -3, -52, -18, -10, -34, 0, -62, -38, -5, -68, -24, -1, -44, -75, 7, -80, -30, -52, -87, 11, -91, -9, -19, -35, -100, -62, -103, -64

Offset: 1

Clark Kimberling, Feb 07 2023

The indexing of diagonals is given in A191360. Conjecture: every integer occurs infinitely many times in this sequence; i.e., every diagonal includes infinitely many primes.

			a(n) = A191360(prime(n)).

Cf. A000040, A035513, A191360, A332938, A360376, A360377, A360378, A360379.

w[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
t = Table[w[n - k + 1, k], {n, 300}, {k, n, 1, -1}];
Map[1 + #[[1]] - 2 #[[2]] &, Most[Reap[NestWhile[# + 1 &, 1,
Length[Sow[FirstPosition[t, Prime[#]]]] > 1 &]][[2]][[1]]]]
(* Peter J. C. Moses, Feb 07 2023 *)

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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A332937 a(n) is the greatest common divisor of the first two terms of row n of the Wythoff array (A035513).

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

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A360379 a(n) = number of the antidiagonal of the Wythoff array (A035513) that includes prime(n).

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A360377 a(n) = number of the row of the Wythoff array (A035513) that includes prime(n).

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A360378 a(n) = number of the column of the Wythoff array (A035513) that includes prime(n).

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A360380 a(n) = number of the diagonal of the Wythoff array, A035513, that includes prime(n). See Comments.

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

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