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A380165 a(n) is the value of the Euler totient function when applied to the largest unitary divisor of n that is an exponentially odd number.

%I A380165 #8 Jan 14 2025 01:52:21
%S A380165 1,1,2,1,4,2,6,4,1,4,10,2,12,6,8,1,16,1,18,4,12,10,22,8,1,12,18,6,28,
%T A380165 8,30,16,20,16,24,1,36,18,24,16,40,12,42,10,4,22,46,2,1,1,32,12,52,18,
%U A380165 40,24,36,28,58,8,60,30,6,1,48,20,66,16,44,24,70,4,72
%N A380165 a(n) is the value of the Euler totient function when applied to the largest unitary divisor of n that is an exponentially odd number.
%H A380165 Amiram Eldar, <a href="/A380165/b380165.txt">Table of n, a(n) for n = 1..10000</a>
%F A380165 a(n) = A000010(A350389(n)).
%F A380165 a(n) >= 1, with equality if and only if n is either a square (A000290) or twice and odd square (A077591 \ {1}).
%F A380165 a(n) <= A000010(n), with equality if and only if n is an exponentially odd number (A268335).
%F A380165 Multiplicative with a(p^e) = (p-1)*p^(e-1) if e is odd, and 1 otherwise.
%F A380165 Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 - 1/p^s + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(3*s-1) + 1/p^(3*s)).
%F A380165 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 2/p^4 + 1/p^5) = 0.50115112192510092436... .
%t A380165 f[p_, e_] := If[OddQ[e], (p-1)*p^(e-1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A380165 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, (f[i, 1]-1)*f[i, 1]^(f[i, 2]-1), 1));}
%Y A380165 Cf. A000010, A000290, A013662, A077591, A268335, A350389, A351569, A365402, A374456, A380164.
%K A380165 nonn,easy,mult
%O A380165 1,3
%A A380165 _Amiram Eldar_, Jan 14 2025