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A380164 a(n) is the value of the Euler totient function when applied to the largest unitary divisor of n that is a square.

%I A380164 #8 Jan 14 2025 01:51:55
%S A380164 1,1,1,2,1,1,1,1,6,1,1,2,1,1,1,8,1,6,1,2,1,1,1,1,20,1,1,2,1,1,1,1,1,1,
%T A380164 1,12,1,1,1,1,1,1,1,2,6,1,1,8,42,20,1,2,1,1,1,1,1,1,1,2,1,1,6,32,1,1,
%U A380164 1,2,1,1,1,6,1,1,20,2,1,1,1,8,54,1,1,2,1,1,1,1,1,6,1,2,1,1,1,1,1,42,6,40
%N A380164 a(n) is the value of the Euler totient function when applied to the largest unitary divisor of n that is a square.
%H A380164 Amiram Eldar, <a href="/A380164/b380164.txt">Table of n, a(n) for n = 1..10000</a>
%F A380164 a(n) = A000010(A350388(n)).
%F A380164 a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
%F A380164 a(n) <= A000010(n), with equality if and only if n is either a square (A000290) or twice an odd square (A077591 \ {1}).
%F A380164 Multiplicative with a(p^e) = (p-1)*p^(e-1) if e is even, and 1 otherwise.
%F A380164 Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s-1) - 1/p^(2*s) - 1/p^(3*s-2) + 1/p^(4*s-1)).
%F A380164 Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(3) * Product_{p prime} (1 + 1/p^(3/2) - 1/p^2 - 1/p^(5/2) - 1/p^3 + 1/p^5) = 1.16404670858123447768... .
%t A380164 f[p_, e_] := If[OddQ[e], 1, (p-1)*p^(e-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A380164 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, 1, (f[i, 1]-1)*f[i, 1]^(f[i, 2]-1)));}
%Y A380164 Cf. A000010, A000290, A002117, A077591, A268335, A350388, A351568, A365401, A380165.
%K A380164 nonn,easy,mult
%O A380164 1,4
%A A380164 _Amiram Eldar_, Jan 14 2025