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A365685 a(n) is the smallest number k such that k*n is an exponentially squarefree number (A209061).

%I A365685 #10 Sep 16 2023 02:23:00
%S A365685 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A365685 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A365685 1,1,1,1,1,1,1,1,1,1,1,2,3,1,1,1,1,1,1
%N A365685 a(n) is the smallest number k such that k*n is an exponentially squarefree number (A209061).
%H A365685 Amiram Eldar, <a href="/A365685/b365685.txt">Table of n, a(n) for n = 1..10000</a>
%F A365685 Multiplicative with a(p^e) = p^A081221(e).
%F A365685 a(n) = A365684(n)/n.
%F A365685 a(n) >= 1, with equality if and only if n is an exponentially squarefree number (A209061).
%F A365685 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} (p^f(k) - p^f(k-1))/p^k) = 1.06562841319..., where f(k) = A081221(k) and f(0) = 0.
%t A365685 f[p_, e_] := Module[{k = e}, While[! SquareFreeQ[k], k++]; p^(k-e)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365685 (PARI) s(e) = {my(k = e); while(!issquarefree(k), k++); k - e;};
%o A365685 a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^s(f[i,2]));}
%Y A365685 Cf. A081221, A209061, A365684.
%K A365685 nonn,easy,mult
%O A365685 1,16
%A A365685 _Amiram Eldar_, Sep 15 2023