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

1,5

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

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).

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 *)

Prog

(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);

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.

Sequence in context: A353644 A007740 A117811 * A297381 A051793 A065371

Adjacent sequences: A349428 A349429 A349430 * A349432 A349433 A349434

Keyword

sign

Author

Antti Karttunen, Nov 17 2021

Status

approved