A319997 a(n) = Sum_{d|n, d is odd} mu(n/d)*d, where mu(n) is Moebius function A008683.

1, -1, 2, 0, 4, -2, 6, 0, 6, -4, 10, 0, 12, -6, 8, 0, 16, -6, 18, 0, 12, -10, 22, 0, 20, -12, 18, 0, 28, -8, 30, 0, 20, -16, 24, 0, 36, -18, 24, 0, 40, -12, 42, 0, 24, -22, 46, 0, 42, -20, 32, 0, 52, -18, 40, 0, 36, -28, 58, 0, 60, -30, 36, 0, 48, -20, 66, 0, 44, -24, 70, 0, 72, -36, 40, 0, 60, -24, 78, 0, 54, -40, 82, 0, 64, -42, 56, 0, 88

Offset: 1

Author

Antti Karttunen, Oct 31 2018

Keywords

• sign
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000010, A000035, A008683, A062570, A319998.

Programs

PARI

A319997(n) = sumdiv(n,d,(d%2)*moebius(n/d)*d);

PARI

A319997(n) = if(n%2, eulerphi(n), if(n%4, -eulerphi(n), 0));

PARI

A319997(n) =  { my(f=factor(n)); prod(i=1,#f~,if(2==f[i,1],-(1==f[i,2]),(f[i,1]-1)*(f[i,1]^(f[i,2]-1)))); };

Formula

a(n) = Sum_{d|n} A000035(d)*A008683(n/d)*d.

a(n) = A000010(n) - A319998(n).

For even n, a(n) = A000010(n) - 2*A000010(n/2); for odd n, a(n) = A000010(n).

a(2n+1) = A000010(2n+1), a(4n+2) = -A000010(4n+2), a(4n) = 0.

Multiplicative with a(2^1) = -1, a(2^e) = 0 for e > 1, and a(p^e) = (p - 1)*p^(e-1) when p is an odd prime.

G.f.: Sum_{k>=1} mu(k)*x^k*(1 + x^(2*k))/(1 - x^(2*k))^2. - Ilya Gutkovskiy, Nov 02 2018

Dirichlet g.f.: zeta(s-1)*(1-2^(1-s))/zeta(s). - R. J. Mathar, Jan 07 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/(2*Pi^2) = 0.151981... . - Amiram Eldar, Nov 12 2022