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A368744 a(n) = Sum_{d|n} (-1)^(d+1)*phi(d), where phi(n) = A000010(n).

%I A368744 #29 Jan 31 2024 08:08:06
%S A368744 1,0,3,-2,5,0,7,-6,9,0,11,-6,13,0,15,-14,17,0,19,-10,21,0,23,-18,25,0,
%T A368744 27,-14,29,0,31,-30,33,0,35,-18,37,0,39,-30,41,0,43,-22,45,0,47,-42,
%U A368744 49,0,51,-26,53,0,55,-42,57,0,59,-30,61,0,63,-62,65,0,67,-34,69,0,71,-54,73,0,75
%N A368744 a(n) = Sum_{d|n} (-1)^(d+1)*phi(d), where phi(n) = A000010(n).
%C A368744 Recall Gauss's identity Sum_{d|n} phi(d) = n.
%C A368744 a(n) is a multiplicative function of n since both (-1)^(n+1) and phi(n) are multiplicative functions of n.
%H A368744 Paolo Xausa, <a href="/A368744/b368744.txt">Table of n, a(n) for n = 1..10000</a>
%F A368744 a(n) = -Sum_{k = 1..n} (-1)^(lcm(k, n)/k) = -Sum_{k = 1..n} (-1)^(n/gcd(k, n)).
%F A368744 a(2*n+1) = 2*n + 1; a(4*n+2) = 0.
%F A368744 Multiplicative: a(2^k) = 2 - 2^k and for odd prime p, a(p^k) = p^k.
%F A368744 Dirichlet g.f.: (1 - 3/2^s)/(1 - 1/2^s) * zeta(s-1).
%F A368744 From _Amiram Eldar_, Jan 31 2024: (Start)
%F A368744 a(n) = (2/A006519(n) - 1) * n.
%F A368744 Sum_{k=1..n} a(k) ~ n^2/6. (End)
%p A368744 with(numtheory): seq( add( (-1)^(d+1)*phi(d), d in divisors(n)), n = 1..75);
%t A368744 A368744[n_] := DivisorSum[n, (-1)^(#+1)*EulerPhi[#]&];
%t A368744 Array[A368744, 100] (* _Paolo Xausa_, Jan 30 2024 *)
%t A368744 a[n_] := (2^(1-IntegerExponent[n, 2]) - 1) * n ; Array[a, 100] (* _Amiram Eldar_, Jan 31 2024 *)
%o A368744 (PARI) a(n) = sumdiv(n, d, (-1)^(d+1)*eulerphi(d)); \\ _Michel Marcus_, Jan 30 2024
%o A368744 (PARI) a(n) = (2/(1<<valuation(n, 2)) - 1) * n; \\ _Amiram Eldar_, Jan 31 2024
%o A368744 (Python)
%o A368744 def A368744(n): return ((n<<1)>>(~n & n-1).bit_length())-n # _Chai Wah Wu_, Jan 30 2024
%Y A368744 Cf. A000010, A006519, A048272, A321543.
%K A368744 sign,mult,easy
%O A368744 1,3
%A A368744 _Peter Bala_, Jan 21 2024