A326937 Dirichlet g.f.: (2^s - 1) / (zeta(s-1) * (2^s - 2)).

1, -1, -3, 0, -5, 3, -7, 0, 0, 5, -11, 0, -13, 7, 15, 0, -17, 0, -19, 0, 21, 11, -23, 0, 0, 13, 0, 0, -29, -15, -31, 0, 33, 17, 35, 0, -37, 19, 39, 0, -41, -21, -43, 0, 0, 23, -47, 0, 0, 0, 51, 0, -53, 0, 55, 0, 57, 29, -59, 0, -61, 31, 0, 0, 65, -33, -67, 0, 69, -35, -71, 0, -73, 37, 0

Offset

1,3

Comments

Dirichlet inverse of A000265.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(1) = 1; a(n) = -Sum_{d|n, d<n} A000265(n/d) * a(d).

a(n) = A055615(n) / A006519(n).

Multiplicative with a(p) = -1 for p = 2 and -p for odd primes p, and a(p^e) = 0 if e > 1. - Amiram Eldar, Dec 02 2020

Mathematica

a[1] = 1; a[n_] := -Sum[(n/d)/2^IntegerExponent[n/d, 2] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 75}]
Table[MoebiusMu[n] n/2^IntegerExponent[n, 2], {n, 1, 75}]
f[2, e_] := -Boole[e == 1]; f[p_, e_] := -Boole[e == 1]*p; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

Prog

(PARI) a(n)={n*moebius(n)/2^valuation(n, 2)} \\ Andrew Howroyd, Oct 25 2019

Crossrefs

Cf. A000265, A006519, A013929 (positions of 0's), A030059 (positions of negative terms), A055615.

Sequence in context: A108500 A322937 A326989 * A336597 A076109 A078788

Adjacent sequences: A326934 A326935 A326936 * A326938 A326939 A326940

Keyword

sign,mult

Author

Ilya Gutkovskiy, Oct 22 2019

Status

approved