A375769 The indices of the terms of A375768 in the Fibonacci sequence.

0, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 4, 3, 2, 4, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 3, 3, 2, 3, 2, 4, 2, 4, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 3, 2, 3, 2, 2, 2, 5, 2, 3, 3, 3, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2

Offset: 1

Author

Amiram Eldar, Aug 27 2024

Keywords

• nonn
• easy

Comments

First differs from A375767 at n = 2448.

Since 1 appears twice in the Fibonacci sequence (1 = Fibonacci(1) = Fibonacci(2)), its index here is chosen to be 2.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000045, A130233, A369939, A375767, A375768.

Programs

Mathematica

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}]; A130233[n_] := Module[{k = 2}, While[Fibonacci[k] <= n, k++]; k-1]; s[n_] := Module[{e = Max[FactorInteger[n][[;; , 2]]]}, If[fibQ[e], A130233[e], Nothing]]; s[1] = 0; Array[s, 100]

PARI

isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
A130233(n) = {my(k = 2); while(fibonacci(k) <= n, k++); k-1;}
lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = vecmax(factor(k)[,2]); if(isfib(e), print1(A130233(e), ", ")));}

Formula

a(n) = A130233(A375768(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (2/zeta(2) + Sum_{k>=3} k * (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k)))) / d = 2.4999593748274972257073..., where d = 1/zeta(4) + Sum_{k>=5} (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k))) = 0.94462177878047854647... is the density of A369939.

If the chosen index for 1 is 1 instead of 2, then the asymptotic mean is (1/zeta(2) + Sum_{k>=3} k * (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k)))) / d = 1.85639269500896710302009... .