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%I #13 Nov 12 2022 05:25:34
%S 1,2,3,2,5,6,7,6,7,10,11,6,13,14,15,10,17,14,19,10,21,22,23,18,23,26,
%T 23,14,29,30,31,22,33,34,35,14,37,38,39,30,41,42,43,22,35,46,47,30,47,
%U 46,51,26,53,46,55,42,57,58,59,30,61,62,49,42,65,66,67,34,69,70,71,42,73,74,69
%N a(n) = Sum_{d|n} (-1)^(bigomega(d) - omega(d)) * phi(n/d).
%C Moebius transform of A327668.
%H Amiram Eldar, <a href="/A333569/b333569.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{k=1..n} (-1)^(bigomega(gcd(n,k)) - omega(gcd(n,k))).
%F a(n) = Sum_{d|n} mu(n/d) * A327668(d).
%F From _Amiram Eldar_, Nov 12 2022: (Start)
%F Multiplicative with a(p) = p, and a(p^e) = (p^e*(p^2+p-2) - 2*(-1)^e*p)/(p*(p+1)) for e>1.
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/5) * Product_{p prime} (1 + 2/p^2) = 0.4381740171... . (End)
%t Table[Sum[(-1)^(PrimeOmega[d] - PrimeNu[d]) EulerPhi[n/d], {d, Divisors[n]}], {n, 1, 75}]
%t Table[Sum[(-1)^(PrimeOmega[GCD[n, k]] - PrimeNu[GCD[n, k]]), {k, 1, n}], {n, 1, 75}]
%t f[p_, e_] := If[e > 1, (p^e*(p^2+p-2) - 2*(-1)^e*p)/(p*(p + 1)), p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100] (* _Amiram Eldar_, Nov 12 2022 *)
%o (PARI) a(n) = sumdiv(n, d, (-1)^(bigomega(d) - omega(d)) * eulerphi(n/d)); \\ _Michel Marcus_, Mar 27 2020
%Y Cf. A000010, A001221, A001222, A005117 (fixed points), A046660, A058026, A074722, A162511, A327666, A327668.
%K nonn,mult
%O 1,2
%A _Ilya Gutkovskiy_, Mar 26 2020
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