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A375766 The maximum exponent in the prime factorization of the numbers whose exponents in their prime factorization are all Fibonacci numbers.

%I A375766 #7 Aug 28 2024 00:56:55
%S A375766 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 A375766 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 A375766 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 A375766 The maximum exponent in the prime factorization of the numbers whose exponents in their prime factorization are all Fibonacci numbers.
%C A375766 First differs from A375768 at n = 2448.
%C A375766 All the terms are Fibonacci numbers by definition.
%H A375766 Amiram Eldar, <a href="/A375766/b375766.txt">Table of n, a(n) for n = 1..10000</a>
%F A375766 a(n) = A051903(A115063(n)).
%F A375766 a(n) = A000045(A375767(n)).
%F A375766 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/zeta(2) + Sum_{k>=3} (Fibonacci(k) * (d(k) - d(k-1)))) / A375274 = 1.52546070254904121983..., where d(k) = Product_{p prime} ((1-1/p)*(1 + Sum_{i=2..k} 1/p^Fibonacci(i))) for k >= 3, and d(2) = 1/zeta(2).
%t A375766 fibQ[n_] := Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}]; s[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, fibQ], Max[e], Nothing]]; s[1] = 0; Array[s, 100]
%o A375766 (PARI) isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
%o A375766 lista(kmax) = {my(e, ans); print1(0, ", "); for(k = 2, kmax, e = factor(k)[,2]; ans = 1; for(i = 1, #e, if(!isfib(e[i]), ans = 0; break)); if(ans, print1(vecmax(e), ", ")));}
%Y A375766 Cf. A000045, A051903, A115063, A375274, A375767, A375768.
%K A375766 nonn,easy
%O A375766 1,4
%A A375766 _Amiram Eldar_, Aug 27 2024