A328639 - OEIS

%I #13 Nov 30 2020 03:58:01

%S 1,-5,-10,5,-26,50,-50,-5,10,130,-122,-50,-170,250,260,5,-290,-50,

%T -362,-130,500,610,-530,50,26,850,-10,-250,-842,-1300,-962,-5,1220,

%U 1450,1300,50,-1370,1810,1700,130,-1682,-2500,-1850,-610,-260,2650,-2210,-50,50,-130

%N Dirichlet g.f.: zeta(2*s) / (zeta(s) * zeta(s-2)).

%C Dirichlet inverse of A065958.

%H Amiram Eldar, <a href="/A328639/b328639.txt">Table of n, a(n) for n = 1..10000</a>

%F a(1) = 1; a(n) = -Sum_{d|n, d<n} A065958(n/d) * a(d).

%F a(n) = Sum_{d|n} lambda(n/d) * mu(d) * d^2, where lambda = A008836 and mu = A008683.

%F Multiplicative with a(p^e) = (-1)^e*(p^2 + 1). - _Amiram Eldar_, Nov 30 2020

%t a[1] = 1; a[n_] := -Sum[DirichletConvolve[j^2, MoebiusMu[j]^2, j, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 50}]

%t Table[DivisorSum[n, LiouvilleLambda[n/#] MoebiusMu[#] #^2 &], {n, 1, 50}]

%t f[p_, e_] := (-1)^e*(p^2 + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Nov 30 2020 *)

%o (PARI) a(n)={sumdiv(n, d, (-1)^bigomega(n/d)*moebius(d)*d^2)} \\ _Andrew Howroyd_, Oct 25 2019

%Y Cf. A008683, A008836, A026424 (positions of negative terms), A046970, A065958, A323363, A328640.

%K sign,mult

%O 1,2

%A _Ilya Gutkovskiy_, Oct 23 2019