A359484 a(n) = n * mu(n) if n is odd, otherwise n * mu(n) - (n/2) * mu(n/2).

1, -3, -3, 2, -5, 9, -7, 0, 0, 15, -11, -6, -13, 21, 15, 0, -17, 0, -19, -10, 21, 33, -23, 0, 0, 39, 0, -14, -29, -45, -31, 0, 33, 51, 35, 0, -37, 57, 39, 0, -41, -63, -43, -22, 0, 69, -47, 0, 0, 0, 51, -26, -53, 0, 55, 0, 57, 87, -59, 30, -61, 93, 0, 0, 65, -99, -67, -34, 69, -105, -71, 0

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

a(n) = A055615(n) if n is odd, otherwise A055615(n) - A055615(n/2).

a(n) is multiplicative with a(2)= -3, a(4)= 2, a(2^e)= 0 if e>2. a(p)= -p, a(p^e)= 0 if e>1, p>2.

a(1) = 1, a(2) = -3; a(n) = -n * Sum_{d|n, d < n} a(d) / d.

G.f. A(x) satisfies x * (1 - x) = Sum_{k>=1} k * A(x^k).

a(n) = n*mu(n)-n*mu(n*2^(n mod 2)/2)*((n+1) mod 2)/2. - Wesley Ivan Hurt, Jun 09 2023

Mathematica

a[n_] := n * MoebiusMu[n] - If[OddQ[n], 0, MoebiusMu[n/2]*n/2]; Array[a, 100] (* Amiram Eldar, May 09 2023 *)

Prog

(PARI) a(n) = n*moebius(n)-if(n%2, 0, n/2*moebius(n/2));

Crossrefs

Partial sums give A359478.

Cf. A008683, A055615, A358276.

Cf. A092673, A359485, A359531.

Sequence in context: A360736 A292527 A359135 * A186111 A186813 A329690

Adjacent sequences: A359481 A359482 A359483 * A359485 A359486 A359487

Keyword

sign,mult

Author

Seiichi Manyama, Mar 31 2023

Status

approved