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

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

1,1

Comments

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

Links

Table of n, a(n) for n=1..86.

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.

Example

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

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}]]

Crossrefs

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

Sequence in context: A304113 A304112 A293446 * A256443 A046068 A166281

Adjacent sequences: A360375 A360376 A360377 * A360379 A360380 A360381

Keyword

nonn,easy

Author

Clark Kimberling, Feb 04 2023

Status

approved