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

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, -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, -4, 0, 0, -4, -2, 0, 4, 0, -2, 0, 0, 0, 0, 0, 4, 0, -2, 0, 2

Offset

1,3

Comments

It seems the nonzero coefficients are powers of 2.

This function is multiplicative with a(p^n) = Re(I^(p^n+1) - I^(p^(n-1)+1)).

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Wikipedia, Dirichlet beta function

Wikipedia, Riemann zeta function

Formula

a(p^n) = (-1)^n*2, for prime p=3 mod 4.

a(p^n) = 0, for prime p=1 mod 4.

a(2) = -1, a(2^n) = 0 for n > 1.

a(n) = -Re(I*Sum_{d|n}(mobius(d)*I^(n/d))).

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.

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.

Sum_{n>=1} mobius(n)/(z^n-I) = Sum_{n >= 1} b(n)/z^n. a(n)=Re(b(n)).

Sum_{n>=1} a(n)/(z^n-1) = z/(z^2+1)

Sum_{d|n} a(d) = A101455(n). - Gevorg Hmayakyan, Dec 27 2017

Example

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.

Maple

a(n):=-Re(I*add(numtheory:-mobius(d)*I^(n/d), d = numtheory:-divisors(n))).

Mathematica

Table[-Re[I *  Sum[MoebiusMu[d] * (I^(n/d)), {d, Divisors[n]}]], {n, 81}] (* Indranil Ghosh, Mar 19 2017 *)

Prog

(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

Crossrefs

Cf. A008683, A101455.

Sequence in context: A337509 A144734 A029361 * A284059 A329767 A356018

Adjacent sequences: A275963 A275964 A275965 * A275967 A275968 A275969

Keyword

sign,mult

Author

Gevorg Hmayakyan, Mar 19 2017

Status

approved