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A386259 Exponent of the highest power of 2 dividing the maximum exponent in the prime factorization of n; a(1) = 0.

%I A386259 #8 Jul 18 2025 04:26:47
%S A386259 0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,2,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,
%T A386259 0,1,0,0,0,0,0,0,0,1,1,0,0,2,1,1,0,1,0,0,0,0,0,0,0,1,0,0,1,1,0,0,0,1,
%U A386259 0,0,0,0,0,0,1,1,0,0,0,2,2,0,0,1,0,0,0
%N A386259 Exponent of the highest power of 2 dividing the maximum exponent in the prime factorization of n; a(1) = 0.
%C A386259 First differs from A386258 at n = 36.
%C A386259 The first occurrence of k = 1, 2, ... is at n = 2^(2^k) = A001146(k).
%C A386259 The asymptotic density of the occurrences of 1 in this sequence is Sum_{k>=0} (1/zeta(4*k+3) - 1/zeta(4*k+2)) = 0.23327987006010127485... .
%H A386259 Amiram Eldar, <a href="/A386259/b386259.txt">Table of n, a(n) for n = 1..10000</a>
%F A386259 a(n) = A007814(A051903(n)) for n >= 2.
%F A386259 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} A001511(k) * (1/zeta(2*k+1)-1/zeta(2*k)) = 0.32062532266909439596... .
%t A386259 a[n_] := IntegerExponent[Max[FactorInteger[n][[;; , 2]]], 2]; a[1] = 0; Array[a, 100]
%o A386259 (PARI) a(n) = if(n == 1, 0, valuation(vecmax(factor(n)[,2]), 2));
%Y A386259 Cf. A001146, A001511, A007814, A051903, A386258.
%K A386259 nonn,easy
%O A386259 1,16
%A A386259 _Amiram Eldar_, Jul 17 2025