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%I #25 Sep 26 2020 04:28:29
%S 1,2,6,0,20,12,42,0,0,40,110,0,156,84,120,0,272,0,342,0,252,220,506,0,
%T 0,312,0,0,812,240,930,0,660,544,840,0,1332,684,936,0,1640,504,1806,0,
%U 0,1012,2162,0,0,0,1632,0,2756,0,2200,0,2052,1624,3422
%N a(n) = n*phi(n)*abs( mobius(n) ).
%C The inverse Mobius transform is b(n>=1) = 1, 3, 7, 3, 21, 21, 43, 3,7, 63, 11, 21,...., multiplicative with b(p^e) = A002061(p), e>=1 (see A119959). - Mathar
%C a(n) > 0 only when n is squarefree. - Alonso del Arte, Dec 20 2011
%H T. D. Noe, <a href="/A202535/b202535.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = A002618(n) *A008966(n).
%F Multiplicative with a(p^e) = (p-1)*p if e=1, a(p^e)=0 if e>1.
%F Dirichlet g.f.: Sum_(n>=1) a(n)/n^s = Product_{primes p} (1-p^(1-s)+p^(2-s)).
%F From _Vaclav Kotesovec_, Jun 24 2020: (Start)
%F Dirichlet g.f.: zeta(s-2)*Product_{primes p} (1 + p^(3-2*s) - p^(4-2*s) - p^(1-s)).
%F Sum_{k=1..n} a(k) ~ c * n^3, where c = A065464/3 = 0.142749835... (End)
%e a(5) = 20 because 5 phi(5) |mu(5)| = 5 * 4 * |(-1)| = 20.
%t Table[n EulerPhi[n] Abs[MoebiusMu[n]], {n, 60}] (* _Alonso del Arte_, Dec 20 2011 *)
%t f[p_, e_] := If[e == 1, (p-1)*p, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 26 2020 *)
%o (PARI) a(n)=n*eulerphi(n)*abs(moebius(n)) \\ _Charles R Greathouse IV_, Dec 20 2011
%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + p^2*X))[n], ", ")) \\ _Vaclav Kotesovec_, Jun 24 2020
%Y Cf. A079579.
%K nonn,mult,easy
%O 1,2
%A _R. J. Mathar_, Dec 20 2011
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