A349452 Dirichlet inverse of A011782, 2^(n-1).

1, -2, -4, -4, -16, -16, -64, -104, -240, -448, -1024, -1904, -4096, -7936, -16256, -32272, -65536, -129888, -262144, -522176, -1048064, -2093056, -4194304, -8379520, -16776960, -33538048, -67106880, -134184704, -268435456, -536801024, -1073741824, -2147352224, -4294959104, -8589672448, -17179867136, -34359197184

Offset: 1

Author

Antti Karttunen, Nov 22 2021

Keywords

• sign

Links

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

Crossrefs

Cf. A011782.

Cf. also A349449, A349450, A349451, A349453 and A349563, A349565, A349567, A349569.

Programs

Mathematica

a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * 2^(n/# - 1) &, # < n &]; Array[a, 36] (* Amiram Eldar, Nov 22 2021 *)

PARI

A011782(n) = (2^(n-1));
memoA349452 = Map();
A349452(n) = if(1==n,1,my(v); if(mapisdefined(memoA349452,n,&v), v, v = -sumdiv(n,d,if(dA011782(n/d)*A349452(d),0)); mapput(memoA349452,n,v); (v)));

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A011782(n/d) * a(d).

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