cp's OEIS Frontend

A380163 a(n) is the value of the Euler totient function when applied to the squarefree part of n.

%I A380163 #9 Jan 14 2025 01:51:51
%S A380163 1,1,2,1,4,2,6,1,1,4,10,2,12,6,8,1,16,1,18,4,12,10,22,2,1,12,2,6,28,8,
%T A380163 30,1,20,16,24,1,36,18,24,4,40,12,42,10,4,22,46,2,1,1,32,12,52,2,40,6,
%U A380163 36,28,58,8,60,30,6,1,48,20,66,16,44,24,70,1,72,36
%N A380163 a(n) is the value of the Euler totient function when applied to the squarefree part of n.
%H A380163 Amiram Eldar, <a href="/A380163/b380163.txt">Table of n, a(n) for n = 1..10000</a>
%F A380163 a(n) = A000010(A007913(n)).
%F A380163 a(n) >= 1, with equality if and only if n is in A028982.
%F A380163 a(n) <= A000010(n), with equality if and only if n is squarefree (A005117).
%F A380163 Multiplicative with a(p^e) = p-1 if e is odd, and 1 otherwise.
%F A380163 Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^s).
%F A380163 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 2/p^2 + 1/p^3) = 0.46350438981962928756...
%t A380163 f[p_, e_] := If[OddQ[e], p-1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A380163 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, f[i, 1]-1, 1));}
%Y A380163 Cf. A000010, A005117, A007913, A013662, A028982, A055076, A367991, A380162.
%K A380163 nonn,easy,mult
%O A380163 1,3
%A A380163 _Amiram Eldar_, Jan 14 2025