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A074722 a(n) = Sum_{d divides n} phi(n/d)*(-1)^bigomega(d).

%I A074722 #33 Aug 28 2023 08:25:52
%S A074722 1,0,1,2,3,0,5,2,5,0,9,2,11,0,3,6,15,0,17,6,5,0,21,2,17,0,13,10,27,0,
%T A074722 29,10,9,0,15,10,35,0,11,6,39,0,41,18,15,0,45,6,37,0,15,22,51,0,27,10,
%U A074722 17,0,57,6,59,0,25,22,33,0,65,30,21,0,69,10,71,0,17,34,45,0,77,18,41,0
%N A074722 a(n) = Sum_{d divides n} phi(n/d)*(-1)^bigomega(d).
%C A074722 a(n) = 0 if and only if n == 2 (mod 4). - _Robert Israel_, Jan 04 2017
%H A074722 Robert Israel, <a href="/A074722/b074722.txt">Table of n, a(n) for n = 1..10000</a>
%F A074722 Multiplicative with a(p^e) = 2*(-1)^(e+1)*((-p)^(e+1)-1)/(p+1)-p^e.
%F A074722 Dirichlet g.f.: zeta(2s)*zeta(s-1)/(zeta(s)^2). - _Benedict W. J. Irwin_, Jul 11 2018
%F A074722 Sum_{k=1..n} a(k) ~ n^2 / 5. - _Vaclav Kotesovec_, Feb 01 2019
%F A074722 a(n) = Sum_{k=1..n} (-1)^bigomega(gcd(n,k)). - _Ilya Gutkovskiy_, Feb 22 2020
%F A074722 Möbius transform of A206369: a(n) = Sum_{d|n} A008683(d) * A206369(n/d). - _Amiram Eldar_, Aug 28 2023
%p A074722 f:= proc(n) uses numtheory; local d;
%p A074722    add(phi(n/d)*(-1)^bigomega(d), d=divisors(n))
%p A074722 end proc:
%p A074722 map(f, [$1..100]); # _Robert Israel_, Jan 04 2017
%t A074722 f[d_] := EulerPhi[n/d] LiouvilleLambda[d]
%t A074722 Table[DivisorSum[n, f], {n, 1, 50}] (* _Benedict W. J. Irwin_, Jul 11 2018 *)
%t A074722 f[p_, e_] := 2*(-1)^(e + 1)*((-p)^(e + 1) - 1)/(p + 1) - p^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 30 2022 *)
%o A074722 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*(-1)^bigomega(d)); \\ _Michel Marcus_, Jul 11 2018
%Y A074722 Cf. A000010, A001222, A008683, A058026, A206369.
%K A074722 nonn,easy,mult
%O A074722 1,4
%A A074722 _Vladeta Jovovic_, Sep 27 2002