login
Moebius transform of negated primes in factorization of n.
3

%I #21 Jan 05 2023 03:19:46

%S 1,-3,-4,6,-6,12,-8,-12,12,18,-12,-24,-14,24,24,24,-18,-36,-20,-36,32,

%T 36,-24,48,30,42,-36,-48,-30,-72,-32,-48,48,54,48,72,-38,60,56,72,-42,

%U -96,-44,-72,-72,72,-48,-96,56,-90,72,-84,-54,108,72,96,80,90,-60,144

%N Moebius transform of negated primes in factorization of n.

%F Multiplicative with a(p^e) = (-1)^e*(p+1)*p^(e-1), e>0. a(1)=1.

%F a(n) = mu(n) * A061019(n) = A008683(n) * A061019(n) = A061020(n) * A007427(n) = A061020(n) * A007428(n) * A000012(n) = A007427(n) * A000012(n) * A061019(n) = A007428(n) * A000005(n) * A061019(n), where operation * denotes Dirichlet convolution. Dirichlet convolution of functions b(n), c(n) is function a(n) = b(n) * c(n) = Sum_{d|n} b(d)*c(n/d).

%F Inverse Moebius transform gives A061019.

%F a(n) = (-1)^A001222(n)*A001615(n).

%F Apparently the Dirichlet inverse of A048250. - _R. J. Mathar_, Jul 15 2010

%F Dirichlet g.f.: zeta(2*s-2)/(zeta(s-1)*zeta(s)). - _Amiram Eldar_, Jan 05 2023

%e a(72) = a(2^3*3^2) = (-1)^3*(2+1)*2^(3-1) * (-1)^2*(3+1)*3^(2-1) = (-12)*12 = -144.

%t f[p_, e_] := (-1)^e*(p + 1)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 05 2023 *)

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (-1)^f[i,2]*(f[i,1]+1)*f[i,1]^(f[i,2]-1));} \\ _Amiram Eldar_, Jan 05 2023

%Y Cf. A061019, A008683, A061020, A007427, A000012, A007428, A000005, A001615, A001222, A000040, A006881, A120944, A000961.

%K sign,mult

%O 1,2

%A _Jaroslav Krizek_, Mar 20 2009