%I #10 Jan 03 2025 09:37:07

%S 1,-3,-4,6,-6,12,-8,-12,3,18,-12,-24,-14,24,24,24,-18,-9,-20,-36,32,

%T 36,-24,48,5,42,0,-48,-30,-72,-32,-48,48,54,48,18,-38,60,56,72,-42,

%U -96,-44,-72,-18,72,-48,-96,7,-15,72,-84,-54,0,72,96,80,90,-60,144,-62,96,-24,96,84,-144,-68,-108,96,-144,-72,-36

%N Dirichlet inverse of A046897, where A046897 is the sum of divisors of n that are not divisible by 4.

%H Antti Karttunen, <a href="/A379497/b379497.txt">Table of n, a(n) for n = 1..20000</a>

%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A046897(n/d) * a(d).

%F From _Amiram Eldar_, Jan 02 2025: (Start)

%F Multiplicative with a(2^e) = -3*(-2)^(e-1), and for an odd prime p, a(p) = -(p+1), a(p^2) = p, and a(p^e) = 0 for e >= 3.

%F Dirichlet g.f.: 1/((1 - 1/4^(s-1)) * zeta(s-1) * zeta(s)). (End)

%t f[p_, e_] := If[e == 1, -p-1, If[e == 2, p, 0]]; f[2, e_] := -3*(-2)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 02 2025 *)

%o (PARI)

%o A046897(n) = if(n<1, 0, sumdiv(n, d, if(d%4, d, 0)));

%o memoA379497 = Map();

%o A379497(n) = if(1==n,1,my(v); if(mapisdefined(memoA379497,n,&v), v, v = -sumdiv(n,d,if(d<n,A046897(n/d)*A379497(d),0)); mapput(memoA379497,n,v); (v)));

%o (PARI) g(p, e) = if(p == 2, -3*(-2)^(e-1), if(e == 1, -p-1, e == 2, p, e > 2, 0));

%o a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; g(p, e));} \\ _Amiram Eldar_, Jan 02 2025

%Y Cf. A046897.

%K sign,mult,easy

%O 1,2

%A _Antti Karttunen_, Jan 02 2025