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A275966 a(n) is the real part of -I*Sum_{d|n}(mobius(d)*I^(n/d)), I=sqrt(-1), mobius(n) is A008683.

%I A275966 #74 Dec 27 2017 11:05:40
%S A275966 1,-1,-2,0,0,2,-2,0,2,0,-2,0,0,2,0,0,0,-2,-2,0,4,2,-2,0,0,0,-2,0,0,0,
%T A275966 -2,0,4,0,0,0,0,2,0,0,0,-4,-2,0,0,2,-2,0,2,0,0,0,0,2,0,0,4,0,-2,0,0,2,
%U A275966 -4,0,0,-4,-2,0,4,0,-2,0,0,0,0,0,4,0,-2,0,2
%N A275966 a(n) is the real part of -I*Sum_{d|n}(mobius(d)*I^(n/d)), I=sqrt(-1), mobius(n) is A008683.
%C A275966 It seems the nonzero coefficients are powers of 2.
%C A275966 This function is multiplicative with a(p^n) = Re(I^(p^n+1) - I^(p^(n-1)+1)).
%H A275966 Charles R Greathouse IV, <a href="/A275966/b275966.txt">Table of n, a(n) for n = 1..10000</a>
%H A275966 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_beta_function">Dirichlet beta function</a>
%H A275966 Wikipedia, <a href="https://en.wikipedia.org/wiki/Riemann_zeta_function">Riemann zeta function</a>
%F A275966 a(p^n) = (-1)^n*2, for prime p=3 mod 4.
%F A275966 a(p^n) = 0, for prime p=1 mod 4.
%F A275966 a(2) = -1, a(2^n) = 0 for n > 1.
%F A275966 a(n) = -Re(I*Sum_{d|n}(mobius(d)*I^(n/d))).
%F A275966 Dirichlet g.f.: Sum_{n >= 1} a(n)/n^s = beta(s)/zeta(s), where beta(s) and zeta(s) are Dirichlet Beta and Riemann zeta functions accordingly.
%F A275966 Sum_{n >= 1} a(n)/n^s = (1-2^(-s))*Product_{p=3 mod 4}(p^s-1)/(p^s+1), where p runs over prime numbers.
%F A275966 Sum_{n>=1} mobius(n)/(z^n-I) = Sum_{n >= 1} b(n)/z^n. a(n)=Re(b(n)).
%F A275966 Sum_{n>=1} a(n)/(z^n-1) = z/(z^2+1)
%F A275966 Sum_{d|n} a(d) = A101455(n). - _Gevorg Hmayakyan_, Dec 27 2017
%e A275966 a(4) = -Re(I*(mobius(1)*I^4 + mobius(2)*I^2 + mobius(4)*I)) = Re((I^4-I^2)*I) = Re(2*I) = 0.
%p A275966 a(n):=-Re(I*add(numtheory:-mobius(d)*I^(n/d), d = numtheory:-divisors(n))).
%t A275966 Table[-Re[I *  Sum[MoebiusMu[d] * (I^(n/d)), {d, Divisors[n]}]], {n, 81}] (* _Indranil Ghosh_, Mar 19 2017 *)
%o A275966 (PARI) a(n)=my(f=factor(n)); prod(i=1,#f~, if(f[i,1]==2, if(f[i,2]==1,-1,0), if(f[i,1]%4==3, 2*(-1)^f[i,2], 0))) \\ _Charles R Greathouse IV_, Mar 22 2017
%Y A275966 Cf. A008683, A101455.
%K A275966 sign,mult
%O A275966 1,3
%A A275966 _Gevorg Hmayakyan_, Mar 19 2017