A355837 - OEIS

%I #17 Dec 29 2022 06:30:08

%S 1,-2,-2,0,-2,4,-2,2,0,4,-2,0,-2,4,4,0,-2,0,-2,0,4,4,-2,-4,0,4,2,0,-2,

%T -8,-2,-2,4,4,4,0,-2,4,4,-4,-2,-8,-2,0,0,4,-2,0,0,0,4,0,-2,-4,4,-4,4,

%U 4,-2,0,-2,4,0,0,4,-8,-2,0,4,-8,-2,0,-2,4,0,0,4,-8,-2,0,0,4,-2,0,4,4,4,-4,-2,0,4,0,4,4,4,4

%N Dirichlet inverse of A322327.

%H Antti Karttunen, <a href="/A355837/b355837.txt">Table of n, a(n) for n = 1..100000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F Multiplicative with a(p^e) = 2 * (e mod 2) * (-1)^((e+1)/2) for prime p and e>0.

%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A322327(n/d) * a(d).

%F Dirichlet g.f.: zeta(4*s)/(zeta(s)^2*zeta(2*s)). - _Amiram Eldar_, Dec 29 2022

%t f[p_, e_] := 2 * (-1)^((e + 1)/2) * Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jul 19 2022 *)

%o (PARI) A355837(n) = factorback(apply(e -> 2*(e%2)*((-1)^((1+e)/2)), factor(n)[, 2]));

%Y Cf. A322327.

%K sign,mult

%O 1,2

%A _Antti Karttunen_, Jul 19 2022, based on _Werner Schulte_'s comment in A322327.