A346612 Moebius transform of A019554.

1, 1, 2, 0, 4, 2, 6, 2, 0, 4, 10, 0, 12, 6, 8, 0, 16, 0, 18, 0, 12, 10, 22, 4, 0, 12, 6, 0, 28, 8, 30, 4, 20, 16, 24, 0, 36, 18, 24, 8, 40, 12, 42, 0, 0, 22, 46, 0, 0, 0, 32, 0, 52, 6, 40, 12, 36, 28, 58, 0, 60, 30, 0, 0, 48, 20, 66, 0, 44, 24, 70, 0, 72, 36, 0, 0, 60, 24, 78, 0, 0

Offset: 1

Author

N. J. A. Sloane, Aug 18 2021

Keywords

• nonn
• mult
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane, Transforms.

Crossrefs

Cf. A002117, A019554, A346613.

Programs

Mathematica

f[p_, e_] := If[EvenQ[e], 0, p^((e + 1)/2) - p^((e - 1)/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 19 2021 *)

PARI

a(n) = sumdiv(n, d, moebius(n/d)*d/core(d, 1)[2]); \\ Michel Marcus, Aug 18 2021

Formula

Multiplicative with a(p^e) = 0 if e is even, and p^((e+1)/2) - p^((e-1)/2) if e is odd. - Amiram Eldar, Aug 19 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = 18*zeta(3)/Pi^4 = 0.222125... . - Amiram Eldar, Nov 18 2022