A360379 -id:A360379 - cp's OEIS frontend

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

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 *)

A360485 a(n) = index of the antidiagonal of the Wythoff array (A035513) that includes n.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 3, 5, 4, 4, 4, 5, 6, 6, 5, 5, 7, 5, 8, 6, 7, 9, 7, 6, 10, 6, 11, 8, 6, 12, 9, 7, 13, 8, 14, 10, 8, 15, 7, 16, 11, 7, 17, 12, 9, 18, 7, 19, 13, 10, 20, 8, 21, 14, 9, 22, 15, 11, 23, 9, 24, 16, 8, 25, 17, 12, 26, 8, 27, 18, 13, 28, 10, 29, 19

Offset: 1

Clark Kimberling, Feb 09 2023

Each m appears exactly m times.

Cf. A035513, A191360, 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]] &, Most[Reap[NestWhileList[# + 1 &, 1,
Length[Sow[FirstPosition[t, #]]] > 1 &]][[2]][[1]]]]
(* Peter J. C. Moses, Feb 08 2023 *)

cp's OEIS Frontend

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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A360485 a(n) = index of the antidiagonal of the Wythoff array (A035513) that includes n.

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Author

Keywords

Comments

Crossrefs

Programs

Mathematica