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A306408 a(n) = Sum_{d|n} (-1)^omega(n/d) * d.

%I A306408 #38 May 01 2020 07:51:40
%S A306408 1,1,2,1,4,2,6,1,5,4,10,2,12,6,8,1,16,5,18,4,12,10,22,2,19,12,14,6,28,
%T A306408 8,30,1,20,16,24,5,36,18,24,4,40,12,42,10,20,22,46,2,41,19,32,12,52,
%U A306408 14,40,6,36,28,58,8,60,30,30,1,48,20,66,16,44,24,70,5,72
%N A306408 a(n) = Sum_{d|n} (-1)^omega(n/d) * d.
%C A306408 If n is squarefree (A005117), then a(n) = A000010(n) where A000010 is the Euler totient function.
%H A306408 Amiram Eldar, <a href="/A306408/b306408.txt">Table of n, a(n) for n = 1..10000</a>
%F A306408 a(n) = n * Sum_{d|n} A076479(d)/d.
%F A306408 Sum_{k=1..n} a(k) ~ A065469 * n*(n+1)/2.
%F A306408 Multiplicative with a(p^e) = p^e - (p^e - 1)/(p-1).
%F A306408 a(n) = Sum_{d|n} mu(d)*tau(d)*sigma(n/d). - _Ridouane Oudra_, Nov 24 2019
%F A306408 a(n) = Sum_{d|n} A058026(d). - _Amiram Eldar_, May 01 2020
%p A306408 with(numtheory): omega := n -> nops(factorset(n)):
%p A306408 seq(add((-1)^omega(n/d)*d, d in divisors(n)), n=1..100); # _Ridouane Oudra_, Nov 24 2019
%t A306408 f[p_, e_] := p^e - (p^e - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 01 2020 *)
%o A306408 (PARI) a(n) = sumdiv(n, d, (-1)^omega(n/d) * d);
%o A306408 (PARI) a(n) = my(f=factor(n)); prod(k=1, #f~, f[k,1]^f[k,2] - (f[k,1]^f[k,2] - 1)/(f[k,1]-1));
%Y A306408 Cf. A058026, A065469, A076479, A206369.
%K A306408 nonn,mult
%O A306408 1,3
%A A306408 _Daniel Suteu_, Apr 05 2019