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URL: https://oeis.org/A360378

⇱ A360378 - OEIS

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A360378
a(n) = number of the column of the Wythoff array (A035513) that includes prime(n).
3
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.
FORMULA
Every prime p has a unique representation p = p(m,k) = F(k+1)*floor(m*phi) + (m-1)*F(k), where F(h) = A000045(h) = h-th Fibonacci number, and phi = (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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 04 2023
STATUS
approved