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
Examples
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.
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
- Wikipedia, Dirichlet beta function
- Wikipedia, Riemann zeta function
Programs
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Maple
a(n):=-Re(I*add(numtheory:-mobius(d)*I^(n/d), d = numtheory:-divisors(n))).
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Mathematica
Table[-Re[I * Sum[MoebiusMu[d] * (I^(n/d)), {d, Divisors[n]}]], {n, 81}] (* Indranil Ghosh, Mar 19 2017 *) -
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
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
Comments