A344373 - OEIS

%I #22 Mar 30 2024 03:00:29

%S 0,-1,0,0,0,-1,0,4,0,-1,0,8,0,-1,0,16,0,3,0,16,0,-1,0,36,0,-1,0,24,0,

%T 15,0,48,0,-1,0,48,0,-1,0,68,0,23,0,40,0,-1,0,112,0,15,0,48,0,27,0,

%U 100,0,-1,0,120,0,-1,0,128,0,39,0,64,0,47,0,180,0,-1,0,72,0,47,0,208,0,-1,0,176,0,-1,0,164,0,99

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

%C The usual OEIS policy is not to include sequences like this where alternate terms are zero; this is an exception.

%C For all n, a(n) >= -1. Equality holds for n = 2 and n = 2*p for an odd prime p.

%H Antti Karttunen, <a href="/A344373/b344373.txt">Table of n, a(n) for n = 1..20000</a>

%H Pruthviraj et al., <a href="https://mathoverflow.net/q/392871">Is it always true that Sum_{i=1..a-1}(-1)^i(a,i) >= -1 ?</a>, 2021.

%F a(n) = -A199084(n) - (-1)^n*n = (-1)^n * (A344371(n) - n).

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

%F a(2*n) = A344372(n) - 2*n = -A106475(n-1).

%F Sum_{k=1..n} a(k) ~ (n^2/4) * ((4/Pi^2)*(log(n) + 2*gamma - 1/2 - 4*log(2)/3 - zeta'(2)/zeta(2)) - 1), where gamma is Euler's constant (A001620). - _Amiram Eldar_, Mar 30 2024

%t Array[Sum[(-1)^k GCD[k, #], {k, # - 1}] &, 90] (* _Michael De Vlieger_, May 16 2021 *)

%o (PARI) A344373(n) = sum(k=1,n-1,((-1)^k)*gcd(k,n)); \\ _Antti Karttunen_, May 16 2021

%Y Cf. A001620, A106475, A199084, A306016, A344371, A344372.

%K sign,easy,look

%O 1,8

%A _Max Alekseyev_, May 16 2021

%E More terms from _Antti Karttunen_, May 16 2021