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A386472 The maximum exponent in the prime factorization of the largest divisor of n whose exponents in its prime factorization are squares.

%I A386472 #7 Jul 23 2025 07:22:03
%S A386472 0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,1,1,
%T A386472 1,1,1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,
%U A386472 1,1,1,1,1,1,1,1,1,1,1,4,4,1,1,1,1,1,1
%N A386472 The maximum exponent in the prime factorization of the largest divisor of n whose exponents in its prime factorization are squares.
%C A386472 All the terms are by definition squares.
%H A386472 Amiram Eldar, <a href="/A386472/b386472.txt">Table of n, a(n) for n = 1..10000</a>
%F A386472 a(n) = A051903(A386469(n)).
%F A386472 a(n) = A048760(A051903(n)).
%F A386472 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=1} (2*k+1)*(1-1/zeta((k+1)^2)) = 1.2383138701540899647042996... .
%t A386472 a[n_] := Floor[Sqrt[Max[FactorInteger[n][[;; , 2]]]]]^2; a[1] = 0; Array[a, 100]
%o A386472 (PARI) a(n) = if(n == 1, 0, sqrtint(vecmax(factor(n)[,2]))^2);
%Y A386472 Cf. A048760, A051903, A386469.
%K A386472 nonn,easy
%O A386472 1,16
%A A386472 _Amiram Eldar_, Jul 23 2025