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%I #21 Nov 30 2022 08:29:02
%S 0,1,3,4,5,6,12,8,10,11,18,12,20,14,24,24,20,18,33,20,30,32,36,24,40,
%T 29,42,33,40,30,72,32,40,48,54,48,55,38,60,56,60,42,96,44,60,66,72,48,
%U 80,55,87,72,70,54,99,72,80,80,90,60
%N a(n) = Sum_{k|n} k*mu_2(n/k).
%C Sum_{k|n} k*mu(n/k) is Euler's phi function. In A181549 mu(n) is replaced by the Moebius function of order 2, mu_2(n), A189021(n).
%H Vincenzo Librandi, <a href="/A181549/b181549.txt">Table of n, a(n) for n = 0..1000</a>
%H Peter Luschny, Sequences related to <a href="http://www.oeis.org/wiki/User:Peter_Luschny/EulerTotient">Euler's totient</a> function.
%F From _Amiram Eldar_, Nov 30 2022: (Start)
%F Multiplicative with a(p)= p + 1, and a(p^e) = p^e + p^(e-1) - p^(e-2) if e > 1.
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 + 1/p^2 - 1/p^4) = 0.7124102278... . (End)
%p A181549 := proc(n) local k; add(k*A189021(n/k),k=divisors(n)) end;
%t mu2[1] = 1; mu2[n_] := Sum[Boole[Divisible[n, d^2]]*MoebiusMu[n/d^2]*MoebiusMu[n/d], {d, Divisors[n]}]; a[n_] := Sum[k*mu2[n/k], {k, Divisors[n]}]; Table[a[n], {n, 0, 59}] (* _Jean-François Alcover_, Feb 05 2014 *)
%t f[p_, e_] := p^e + p^(e - 1) - If[e > 1, p^(e - 2), 0]; a[0] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* _Amiram Eldar_, Nov 30 2022 *)
%o (PARI) a(n) = if(n == 0, 0, my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] + f[i,1]^(f[i,2]-1) - if(f[i,2] > 1, f[i,1]^(f[i,2]-2), 0))); \\ _Amiram Eldar_, Nov 30 2022
%Y Cf. A000010, A008683, A181550, A181552, A181553, A189021.
%K nonn,mult
%O 0,3
%A _Peter Luschny_, Oct 30 2010
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