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A375669 The maximum exponent in the prime factorization of the largest odd divisor of n.

%I A375669 #7 Aug 23 2024 10:43:23
%S A375669 0,0,1,0,1,1,1,0,2,1,1,1,1,1,1,0,1,2,1,1,1,1,1,1,2,1,3,1,1,1,1,0,1,1,
%T A375669 1,2,1,1,1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,3,1,1,1,1,1,1,1,1,2,0,1,1,1,1,
%U A375669 1,1,1,2,1,1,2,1,1,1,1,1,4,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,2,2,1,1,1,1,1
%N A375669 The maximum exponent in the prime factorization of the largest odd divisor of n.
%C A375669 The largest exponent among the exponents of the odd primes in the prime factorization of n.
%H A375669 Amiram Eldar, <a href="/A375669/b375669.txt">Table of n, a(n) for n = 1..10000</a>
%H A375669 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A375669 a(n) = A051903(A000265(n)).
%F A375669 a(n) = 0 if and only if n is a power of 2 (A000079).
%F A375669 a(n) = 1 if and only if n is in A122132 \ A000079.
%F A375669 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * d(k) = 1.25979668632898014495... , where d(k) is the asymptotic density of the occurrences of k in this sequence: d(1) = 4/(3*zeta(2)), and d(k) = (1/zeta(k+1)) / (1-1/2^(k+1)) - (1/zeta(k)) / (1-1/2^k) for k >= 2.
%t A375669 a[n_] := Module[{o = n / 2^IntegerExponent[n, 2]}, If[o == 1, 0, Max[FactorInteger[o][[;;, 2]]]]]; Array[a, 100]
%o A375669 (PARI) a(n) = {my(o = n >> valuation(n, 2)); if(o == 1, 0, vecmax(factor(o)[,2]));}
%Y A375669 Cf. A000079, A000265, A051903, A122132, A375670.
%K A375669 nonn,easy
%O A375669 1,9
%A A375669 _Amiram Eldar_, Aug 23 2024