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A375033 The maximum even exponent in the prime factorization of n, or 0 if no such exponent exists.

%I A375033 #12 Aug 17 2024 09:16:30
%S A375033 0,0,0,2,0,0,0,0,2,0,0,2,0,0,0,4,0,2,0,2,0,0,0,0,2,0,0,2,0,0,0,0,0,0,
%T A375033 0,2,0,0,0,0,0,0,0,2,2,0,0,4,2,2,0,2,0,0,0,0,0,0,0,2,0,0,2,6,0,0,0,2,
%U A375033 0,0,0,2,0,0,2,2,0,0,0,4,4,0,0,2,0,0,0,0,0,2,0,2,0,0,0,0,0,2,2,2,0,0,0,0,0
%N A375033 The maximum even exponent in the prime factorization of n, or 0 if no such exponent exists.
%C A375033 First differs from A350386 at n = 36.
%C A375033 The asymptotic density of the occurrences of 0's is d(0) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463; the asymptotic density of the exponentially odd numbers, A268335).
%C A375033 The asymptotic density of the occurrences of 2*k, for k = 1, 2, ..., is d(k) = Product_{p prime} (1 - 1/(p^(2*k+1)*(p+1))) - Product_{p prime} (1 - 1/(p^(2*k-1)*(p+1))).
%C A375033 For example, the asymptotic density of the occurrences of 2's is d(1) = Product_{p prime} (1 - 1/(p^3*(p+1))) - Product_{p prime}(1 - 1/(p*(p+1))) = 0.243291... (the asymptotic density of A375031).
%H A375033 Amiram Eldar, <a href="/A375033/b375033.txt">Table of n, a(n) for n = 1..10000</a>
%H A375033 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A375033 max(a(n), A375032(n)) = A051903(n).
%F A375033 a(n) = 0 if and only if n is an exponentially odd number (A268335).
%F A375033 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (2*k) * d(k) = 0.72584606502990528747..., where d(k) is defined in the Comments section above.
%F A375033 a(n) = A051903(A350388(n)). - _Amiram Eldar_, Aug 17 2024
%t A375033 a[n_] := Max[0, Max[Select[FactorInteger[n][[;; , 2]], EvenQ]]]; a[1] = 0; Array[a, 100]
%o A375033 (PARI) a(n) = {my(e = select(x -> !(x % 2), factor(n)[,2])); if(#e == 0, 0, vecmax(e));}
%Y A375033 Cf. A051903, A065463, A162642, A268335, A350386, A350388, A375031, A375032.
%K A375033 nonn,easy
%O A375033 1,4
%A A375033 _Amiram Eldar_, Jul 28 2024