A365685 - OEIS

%I #10 Sep 16 2023 02:23:00

%S 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 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 1,1,1,1,1,1,1,1,1,1,1,2,3,1,1,1,1,1,1

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

%H Amiram Eldar, <a href="/A365685/b365685.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = p^A081221(e).

%F a(n) = A365684(n)/n.

%F a(n) >= 1, with equality if and only if n is an exponentially squarefree number (A209061).

%F 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 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 (PARI) s(e) = {my(k = e); while(!issquarefree(k), k++); k - e;};

%o a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^s(f[i,2]));}

%Y Cf. A081221, A209061, A365684.

%K nonn,easy,mult

%O 1,16

%A _Amiram Eldar_, Sep 15 2023