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A380162 a(n) is the value of the Euler totient function when applied to the largest square dividing n.

%I A380162 #9 Jan 14 2025 01:51:50
%S A380162 1,1,1,2,1,1,1,2,6,1,1,2,1,1,1,8,1,6,1,2,1,1,1,2,20,1,6,2,1,1,1,8,1,1,
%T A380162 1,12,1,1,1,2,1,1,1,2,6,1,1,8,42,20,1,2,1,6,1,2,1,1,1,2,1,1,6,32,1,1,
%U A380162 1,2,1,1,1,12,1,1,20,2,1,1,1,8,54,1,1,2,1
%N A380162 a(n) is the value of the Euler totient function when applied to the largest square dividing n.
%H A380162 Amiram Eldar, <a href="/A380162/b380162.txt">Table of n, a(n) for n = 1..10000</a>
%F A380162 a(n) = A000010(A008833(n)).
%F A380162 a(n) >= 1, with equality if and only if n is squarefree (A005117).
%F A380162 a(n) <= A000010(n), with equality if and only if n is either a square (A000290) or twice an odd square (A077591 \ {1}).
%F A380162 Multiplicative with a(p) = 1, and a(p^e) = (p-1)*p^(2*floor(e/2)-1) if e >= 2.
%F A380162 Dirichlet g.f.: zeta(s) * zeta(2*s-2) / (zeta(2*s-1) * zeta(2*s)).
%F A380162 Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(3/2)/(zeta(2)*zeta(3)) = 1.32118019580177760682... .
%t A380162 f[p_, e_] := If[e == 1, 1, (p-1)*p^(2*Floor[e/2]-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A380162 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, (f[i, 1]-1) * f[i, 1]^(2*(f[i, 2]\2)-1)));}
%Y A380162 Cf. A000010, A000290, A005117, A008833, A077591, A365331, A365332, A380163.
%Y A380162 Cf. A002117, A013661, A078434.
%K A380162 nonn,easy,mult
%O A380162 1,4
%A A380162 _Amiram Eldar_, Jan 13 2025