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A386468 The maximum exponent in the prime factorization of the largest exponentially squarefree divisor of n.

%I A386468 #8 Jul 23 2025 07:21:54
%S A386468 0,1,1,2,1,1,1,3,2,1,1,2,1,1,1,3,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,5,1,1,
%T A386468 1,2,1,1,1,3,1,1,1,2,2,1,1,3,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,6,1,1,1,2,
%U A386468 1,1,1,3,1,1,2,2,1,1,1,3,3,1,1,2,1,1,1
%N A386468 The maximum exponent in the prime factorization of the largest exponentially squarefree divisor of n.
%C A386468 First differs from A375428 at n = 64.
%C A386468 Differs from A368105 at n = 1, 36, 64, 72, 100, ... .
%C A386468 Except for a(1), all the terms are by definition squarefree numbers.
%H A386468 Amiram Eldar, <a href="/A386468/b386468.txt">Table of n, a(n) for n = 1..10000</a>
%F A386468 a(n) = A051903(A365683(n)).
%F A386468 a(n) = A070321(A051903(n)) for n >= 2.
%F A386468 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=2} A378085(k-1)*(1-1/zeta(k)) = 1.66055078443790141429... .
%t A386468 a[n_] := Module[{k = Max[FactorInteger[n][[;; , 2]]]}, While[! SquareFreeQ[k], k--]; k]; a[1] = 0; Array[a, 100]
%o A386468 (PARI) a(n) = if(n == 1, 0, my(k = vecmax(factor(n)[,2])); while(!issquarefree(k), k--); k);
%Y A386468 Cf. A005117, A051903, A070321, A365683, A368105, A375428, A378085.
%K A386468 nonn,easy
%O A386468 1,4
%A A386468 _Amiram Eldar_, Jul 22 2025