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A386469 The largest divisor of n whose exponents in its prime factorization are squares.

%I A386469 #7 Jul 23 2025 07:21:30
%S A386469 1,2,3,2,5,6,7,2,3,10,11,6,13,14,15,16,17,6,19,10,21,22,23,6,5,26,3,
%T A386469 14,29,30,31,16,33,34,35,6,37,38,39,10,41,42,43,22,15,46,47,48,7,10,
%U A386469 51,26,53,6,55,14,57,58,59,30,61,62,21,16,65,66,67,34,69,70
%N A386469 The largest divisor of n whose exponents in its prime factorization are squares.
%C A386469 The largest term in A197680 that divides n.
%C A386469 The number of these divisors is A386470(n) and their sum is A386471(n).
%H A386469 Amiram Eldar, <a href="/A386469/b386469.txt">Table of n, a(n) for n = 1..10000</a>
%F A386469 Multiplicative with a(p^e) = p^A048760(e).
%F A386469 a(n) <= n, with equality if and only if n is in A197680.
%F A386469 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} Sum_{k>=2} (1/p^(k^2-1) - 1/p^(k^2-2)) = 0.74491327356409794092... .
%t A386469 f[p_, e_] := p^(Floor[Sqrt[e]]^2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A386469 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(sqrtint(f[i, 2])^2)); }
%Y A386469 Cf. A048760, A197680, A386470, A386471.
%Y A386469 Similar sequences: A008833, A350390, A365683.
%K A386469 nonn,easy,mult
%O A386469 1,2
%A A386469 _Amiram Eldar_, Jul 22 2025