A349431 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A055615 (Dirichlet inverse of n).

1, -1, -1, -1, -2, 1, -3, -1, -1, 2, -5, 1, -6, 3, 4, -1, -8, 1, -9, 2, 6, 5, -11, 1, -2, 6, -1, 3, -14, -4, -15, -1, 10, 8, 12, 1, -18, 9, 12, 2, -20, -6, -21, 5, 4, 11, -23, 1, -3, 2, 16, 6, -26, 1, 20, 3, 18, 14, -29, -4, -30, 15, 6, -1, 24, -10, -33, 8, 22, -12, -35, 1, -36, 18, 4, 9, 30, -12, -39, 2, -1, 20

Offset: 1

Author

Antti Karttunen, Nov 17 2021

Keywords

• sign

Comments

Dirichlet convolution of this sequence with A000010 gives A349136, which also proves the formula involving A023900.

Convolution with A000203 gives A349371.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Sequence A297381 negated.

Cf. A003602, A023900, A055615, A297381, A349432 (Dirichlet inverse), A349433 (sum with it).

Cf. also A000010, A000203, A349136, A349371, and also A349444, A349447.

Programs

Mathematica

k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, # * MoebiusMu [#] * k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

PARI

A003602(n) = (1+(n>>valuation(n,2)))/2;
A055615(n) = (n*moebius(n));
A349431(n) = sumdiv(n,d,A003602(n/d)*A055615(d));

PARI

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A349431(n) = if(!bitand(n,n-1),A023900(n),A023900(n)/2);

Formula

a(n) = Sum_{d|n} A003602(n/d) * A055615(d).

a(n) = A023900(n) when n is a power of 2, and a(n) = A023900(n)/2 for all other numbers.

a(n) = -A297381(n).