A349453 Dirichlet inverse of A133494, 3^(n-1).

1, -3, -9, -18, -81, -189, -729, -2052, -6480, -19197, -59049, -175446, -531441, -1589949, -4781511, -14335704, -43046721, -129097152, -387420489, -1162141182, -3486771279, -10459998909, -31381059609, -94142073420, -282429529920, -847285420797, -2541865710960, -7625587899366, -22876792454961, -68630348286531

Offset: 1

Author

Antti Karttunen, Nov 22 2021

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1001

Crossrefs

Cf. A133494.

Cf. also A349449, A349450, A349451, A349452, A349568.

Programs

Mathematica

a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * 3^(n/# - 1) &, # < n &]; Array[a, 30] (* Amiram Eldar, Nov 22 2021 *)

PARI

A133494(n) = max(1, 3^(n-1));
memoA349453 = Map();
A349453(n) = if(1==n,1,my(v); if(mapisdefined(memoA349453,n,&v), v, v = -sumdiv(n,d,if(dA133494(n/d)*A349453(d),0)); mapput(memoA349453,n,v); (v)));

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A133494(n/d) * a(d).

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} 3^(k-1) * A(x^k). - Ilya Gutkovskiy, Feb 23 2022