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A368711 The maximal exponent in the prime factorization of the exponentially odd numbers (A268335).

%I A368711 #11 Apr 26 2025 21:36:10
%S A368711 0,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,3,1,3,1,1,1,5,1,1,1,1,1,1,3,1,1,1,
%T A368711 1,1,1,1,3,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,
%U A368711 5,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,3,1
%N A368711 The maximal exponent in the prime factorization of the exponentially odd numbers (A268335).
%C A368711 Differs from A368472 at n = 1, 154, 610, 707, 762, ... .
%H A368711 Amiram Eldar, <a href="/A368711/b368711.txt">Table of n, a(n) for n = 1..10000</a>
%F A368711 a(n) = A051903(A268335(n)).
%F A368711 a(n) is odd for n >= 2.
%F A368711 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + 2 * Sum_{k>=1} (1 - Product_{p prime} (1 - 1/(p^(2*k-1)*(p^2+p-1)))) = 1.34877064483679975726... .
%t A368711 f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, OddQ], Max @@ e, Nothing]]; f[1] = 0; Array[f, 150]
%o A368711 (PARI) lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = factor(k)[,2]; if(vecprod(e)%2, print1(vecmax(e), ", ")));}
%Y A368711 Cf. A033150, A051903, A268335, A368472.
%Y A368711 Similar sequences: A368710, A368712, A368713.
%K A368711 nonn,easy
%O A368711 1,7
%A A368711 _Amiram Eldar_, Jan 04 2024