A130054 Inverse Moebius transform of A023900.

1, 0, -1, -1, -3, 0, -5, -2, -3, 0, -9, 1, -11, 0, 3, -3, -15, 0, -17, 3, 5, 0, -21, 2, -7, 0, -5, 5, -27, 0, -29, -4, 9, 0, 15, 3, -35, 0, 11, 6, -39, 0, -41, 9, 9, 0, -45, 3, -11, 0, 15, 11, -51, 0, 27, 10, 17, 0, -57, -3, -59, 0, 15, -5, 33, 0, -65, 15, 21

Offset: 1

Author

Gary W. Adamson, May 04 2007

Keywords

• sign
• mult
• easy

Comments

Multiplicative because A023900 is. - Andrew Howroyd, Aug 03 2018

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)

Crossrefs

Row sums of A129691.

Cf. A126988, A000005, A023900, A062570, A001511, A018804, A000203, A007431.

Programs

Magma

[&+[d*MoebiusMu(d)*NumberOfDivisors(n div d):d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 17 2019

Maple

with(numtheory): seq(add(d*mobius(d)*tau(n/d), d in divisors(n)), n=1..60); # Ridouane Oudra, Nov 17 2019

Mathematica

b[n_] := Sum[d MoebiusMu[d], {d, Divisors[n]}];
a[n_] := Sum[b[n/d], {d, Divisors[n]}];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019, from PARI *)
f[p_, e_] := 1-(p-1)*e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 23 2020 *)

PARI

\\ here b(n) is A023900
b(n)={sumdivmult(n, d, d*moebius(d))}
a(n)={sumdiv(n, d, b(n/d))} \\ Andrew Howroyd, Aug 03 2018

Formula

A126988 * A130054 = d(n), A000005: (1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ...).

a(n) = Sum_{d|n} A023900(n/d). - Andrew Howroyd, Aug 03 2018

a(n) = Sum_{d|n} d*mu(d)*tau(n/d). - Ridouane Oudra, Nov 17 2019

From Werner Schulte, Sep 06 2020: (Start)

Multiplicative with a(p^e) = 1 - (p-1) * e for prime p and e >= 0.

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

Dirichlet convolution with A062570 equals A001511.

Dirichlet convolution with A018804 equals A000203.

Dirichlet inverse of A007431. (End)

a(n) = 1 - Sum_{k=1..n-1} a(gcd(n,k)). - Ilya Gutkovskiy, Nov 06 2020

Extensions

Name changed and terms a(11) and beyond from Andrew Howroyd, Aug 03 2018