cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Clark Kimberling, Feb 05 2023

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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 *)

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

Views

Author

Clark Kimberling, Feb 04 2023

Keywords

Comments

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

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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}]]

Formula

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

Views

Author

Clark Kimberling, Feb 07 2023

Keywords

Comments

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.

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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 *)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.