A317673 Moebius transform of A129502.

1, 2, -1, 3, -1, -2, -1, 4, 0, -2, -1, -3, -1, -2, 1, 5, -1, 0, -1, -3, 1, -2, -1, -4, 0, -2, 0, -3, -1, 2, -1, 6, 1, -2, 1, 0, -1, -2, 1, -4, -1, 2, -1, -3, 0, -2, -1, -5, 0, 0, 1, -3, -1, 0, 1, -4, 1, -2, -1, 3, -1, -2, 0, 7, 1, 2, -1, -3, 1, 2, -1, 0, -1

Offset: 1

Author

Andrew Howroyd, Aug 03 2018

Keywords

• sign
• easy
• mult

Comments

Dirichlet convolution of A209635 and A209229.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1024

Crossrefs

First column of A129501.

Cf. A129502, A209229, A209635 (Moebius transform of A104117).

Programs

Mathematica

a[n_] := Module[{e}, Sum[e = IntegerExponent[d, 2]; If[d == 2^e, MoebiusMu[n/d] Binomial[2 + e, 2], 0], {d, Divisors[n]}]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 24 2019, from PARI *)
f[p_, e_] := If[e == 1, -1, 0]; f[2, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 70] (* Amiram Eldar, Aug 28 2023 *)

PARI

a(n)={sumdiv(n, d,  my(e=valuation(d, 2)); if(d==1<
				

Formula

Multiplicative with a(2^e) = e+1, and if p is an odd prime, a(p) = -1 and a(p^e) = 0 for e >= 2. - Amiram Eldar, Aug 28 2023