cp's OEIS Frontend

A380160 a(n) is the value of the Euler totient function when applied to the powerful part of n.

%I A380160 #10 Jan 14 2025 01:52:16
%S A380160 1,1,1,2,1,1,1,4,6,1,1,2,1,1,1,8,1,6,1,2,1,1,1,4,20,1,18,2,1,1,1,16,1,
%T A380160 1,1,12,1,1,1,4,1,1,1,2,6,1,1,8,42,20,1,2,1,18,1,4,1,1,1,2,1,1,6,32,1,
%U A380160 1,1,2,1,1,1,24,1,1,20,2,1,1,1,8,54,1,1,2
%N A380160 a(n) is the value of the Euler totient function when applied to the powerful part of n.
%H A380160 Amiram Eldar, <a href="/A380160/b380160.txt">Table of n, a(n) for n = 1..10000</a>
%F A380160 a(n) = A000010(A057521(n)).
%F A380160 a(n) >= 1, with equality if and only if n is squarefree (A005117).
%F A380160 a(n) <= A000010(n), with equality if and only if n is powerful (A001694).
%F A380160 Multiplicative with a(p) = 1, and a(p^e) = (p-1)*p^(e-1) if e >= 2.
%F A380160 Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(s-1) + 1/p^(2*s-2) - 2/p^(2*s-1)).
%F A380160 Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = Product_{p prime} (1 + 2/p^(3/2) - 1/p^2 - 2/p^(5/2)) = 1.96428740396979919886... .
%t A380160 f[p_, e_] := If[e == 1, 1, (p-1)*p^(e-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A380160 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, (f[i, 1]-1)*f[i, 1]^(f[i, 2]-1)));}
%Y A380160 Cf. A000010, A001694, A005117, A057521, A295294, A295295, A323333, A349379, A357669, A380161.
%K A380160 nonn,easy,mult
%O A380160 1,4
%A A380160 _Amiram Eldar_, Jan 13 2025