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

%I A375769 #15 Apr 13 2025 19:57:18
%S A375769 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,
%T A375769 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,
%U A375769 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
%N A375769 The indices of the terms of A375768 in the Fibonacci sequence.
%C A375769 First differs from A375767 at n = 2448.
%C A375769 Since 1 appears twice in the Fibonacci sequence (1 = Fibonacci(1) = Fibonacci(2)), its index here is chosen to be 2.
%H A375769 Amiram Eldar, <a href="/A375769/b375769.txt">Table of n, a(n) for n = 1..10000</a>
%F A375769 a(n) = A130233(A375768(n)).
%F A375769 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.
%F A375769 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... .
%t A375769 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]
%o A375769 (PARI) isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
%o A375769 A130233(n) = {my(k = 2); while(fibonacci(k) <= n, k++); k-1;}
%o A375769 lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = vecmax(factor(k)[,2]); if(isfib(e), print1(A130233(e), ", ")));}
%Y A375769 Cf. A000045, A130233, A369939, A375767, A375768.
%K A375769 nonn,easy
%O A375769 1,2
%A A375769 _Amiram Eldar_, Aug 27 2024