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A375768 The maximum exponent in the prime factorization of the numbers whose maximum exponent in their prime factorization is a Fibonacci number.

%I A375768 #10 Mar 30 2025 04:51:51
%S A375768 0,1,1,2,1,1,1,3,2,1,1,2,1,1,1,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,5,1,1,1,
%T A375768 2,1,1,1,3,1,1,1,2,2,1,1,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,1,1,1,2,1,1,1,
%U A375768 3,1,1,2,2,1,1,1,1,1,2,1,1,1,3,1,2,1,2,1,1,1,5,1,2,2,2,1,1,1,3,1,1,1,3,1,1
%N A375768 The maximum exponent in the prime factorization of the numbers whose maximum exponent in their prime factorization is a Fibonacci number.
%C A375768 First differs from A375766 at n = 2448.
%C A375768 All the terms are Fibonacci numbers by definition.
%H A375768 Amiram Eldar, <a href="/A375768/b375768.txt">Table of n, a(n) for n = 1..10000</a>
%F A375768 a(n) = A051903(A369939(n)).
%F A375768 a(n) = A000045(A375769(n)).
%F A375768 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/zeta(2) + Sum_{k>=3} Fibonacci(k) * (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k)))) / d = 1.52660290991620063268..., where d = 1/zeta(4) + Sum_{k>=5} (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k))) = 0.94462177878047854647... is the density of A369939.
%t A375768 fibQ[n_] := Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}]; s[n_] := Module[{e = Max[FactorInteger[n][[;; , 2]]]}, If[fibQ[e], e, Nothing]]; s[1] = 0; Array[s, 100]
%o A375768 (PARI) isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
%o A375768 lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = vecmax(factor(k)[,2]); if(isfib(e), print1(e, ", ")));}
%Y A375768 Cf. A000045, A051903, A369939, A375766, A375769.
%K A375768 nonn,easy
%O A375768 1,4
%A A375768 _Amiram Eldar_, Aug 27 2024