A368744 - OEIS

%I #29 Jan 31 2024 08:08:06

%S 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 27,-14,29,0,31,-30,33,0,35,-18,37,0,39,-30,41,0,43,-22,45,0,47,-42,

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

%C Recall Gauss's identity Sum_{d|n} phi(d) = n.

%C a(n) is a multiplicative function of n since both (-1)^(n+1) and phi(n) are multiplicative functions of n.

%H Paolo Xausa, <a href="/A368744/b368744.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = -Sum_{k = 1..n} (-1)^(lcm(k, n)/k) = -Sum_{k = 1..n} (-1)^(n/gcd(k, n)).

%F a(2*n+1) = 2*n + 1; a(4*n+2) = 0.

%F Multiplicative: a(2^k) = 2 - 2^k and for odd prime p, a(p^k) = p^k.

%F Dirichlet g.f.: (1 - 3/2^s)/(1 - 1/2^s) * zeta(s-1).

%F From _Amiram Eldar_, Jan 31 2024: (Start)

%F a(n) = (2/A006519(n) - 1) * n.

%F Sum_{k=1..n} a(k) ~ n^2/6. (End)

%p with(numtheory): seq( add( (-1)^(d+1)*phi(d), d in divisors(n)), n = 1..75);

%t A368744[n_] := DivisorSum[n, (-1)^(#+1)*EulerPhi[#]&];

%t Array[A368744, 100] (* _Paolo Xausa_, Jan 30 2024 *)

%t a[n_] := (2^(1-IntegerExponent[n, 2]) - 1) * n ; Array[a, 100] (* _Amiram Eldar_, Jan 31 2024 *)

%o (PARI) a(n) = sumdiv(n, d, (-1)^(d+1)*eulerphi(d)); \\ _Michel Marcus_, Jan 30 2024

%o (PARI) a(n) = (2/(1<<valuation(n, 2)) - 1) * n; \\ _Amiram Eldar_, Jan 31 2024

%o (Python)

%o def A368744(n): return ((n<<1)>>(~n & n-1).bit_length())-n # _Chai Wah Wu_, Jan 30 2024

%Y Cf. A000010, A006519, A048272, A321543.

%K sign,mult,easy

%O 1,3

%A _Peter Bala_, Jan 21 2024