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

Author

Clark Kimberling, Feb 04 2023

Keywords

• nonn
• easy

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.