A349569 Dirichlet convolution of A000027 (identity function) with A349452 (Dirichlet inverse of A011782, 2^(n-1)).

1, 0, -1, -4, -11, -24, -57, -112, -243, -480, -1013, -1964, -4083, -8064, -16309, -32496, -65519, -130440, -262125, -523156, -1048263, -2095104, -4194281, -8383760, -16777015, -33546240, -67107609, -134200860, -268435427, -536835096, -1073741793, -2147417216, -4294962187, -8589803520, -17179867533, -34359463812

Offset: 1

Author

Antti Karttunen, Nov 22 2021

Keywords

• sign

Comments

Dirichlet convolution with A034729 gives sigma, A000203, and convolution with A034738 gives A018804.

Links

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

Crossrefs

Cf. A000027, A011782, A349452, A349570 (Dirichlet inverse).

Cf. also A000203, A018804, A034729, A034738, A349563, A349565, A349567.

Programs

Mathematica

s[1] = 1; s[n_] := s[n] = -DivisorSum[n, s[#]*2^(n/# - 1) &, # < n &]; a[n_] := DivisorSum[n, # * s[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)));
A349569(n) = sumdiv(n,d,d * A349452(n/d));

Formula

a(n) = Sum_{d|n} d * A349452(n/d).