|
%I #25 Nov 12 2022 05:25:20
%S 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,
%T 18,0,28,-8,30,0,20,-16,24,0,36,-18,24,0,40,-12,42,0,24,-22,46,0,42,
%U -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
%N a(n) = Sum_{d|n, d is odd} mu(n/d)*d, where mu(n) is Moebius function A008683.
%H Antti Karttunen, <a href="/A319997/b319997.txt">Table of n, a(n) for n = 1..20000</a>
%F a(n) = Sum_{d|n} A000035(d)*A008683(n/d)*d.
%F a(n) = A000010(n) - A319998(n).
%F For even n, a(n) = A000010(n) - 2*A000010(n/2); for odd n, a(n) = A000010(n).
%F a(2n+1) = A000010(2n+1), a(4n+2) = -A000010(4n+2), a(4n) = 0.
%F 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.
%F G.f.: Sum_{k>=1} mu(k)*x^k*(1 + x^(2*k))/(1 - x^(2*k))^2. - _Ilya Gutkovskiy_, Nov 02 2018
%F Dirichlet g.f.: zeta(s-1)*(1-2^(1-s))/zeta(s). - _R. J. Mathar_, Jan 07 2021
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/(2*Pi^2) = 0.151981... . - _Amiram Eldar_, Nov 12 2022
%o (PARI) A319997(n) = sumdiv(n,d,(d%2)*moebius(n/d)*d);
%o (PARI) A319997(n) = if(n%2, eulerphi(n), if(n%4, -eulerphi(n), 0));
%o (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)))); };
%Y Cf. A000010, A000035, A008683, A062570, A319998.
%K sign,mult
%O 1,3
%A _Antti Karttunen_, Oct 31 2018
|
