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A360377
a(n) = number of the row of the Wythoff array (A035513) that includes prime(n).
3
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
1,4
COMMENTS
Conjecture: every primitive row number, as defined in A332938, occurs infinitely many times in this sequence.
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 m such that prime(n) = p(m,k) for some k.
EXAMPLE
The 10th prime is 29, which occurs in row 7, so a(10) = 2.
MATHEMATICA
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}]]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 04 2023
STATUS
approved