A363012 a(n) = Sum_{d|n} d*tau(d)*phi(d)*mu(n/d).

1, 3, 11, 20, 39, 33, 83, 104, 150, 117, 219, 220, 311, 249, 429, 512, 543, 450, 683, 780, 913, 657, 1011, 1144, 1460, 933, 1782, 1660, 1623, 1287, 1859, 2432, 2409, 1629, 3237, 3000, 2663, 2049, 3421, 4056, 3279, 2739, 3611, 4380, 5850, 3033, 4323, 5632, 6090, 4380

Offset: 1

Author

Wesley Ivan Hurt, May 12 2023

Keywords

• easy
• nonn
• mult

Comments

Möbius transform of n*tau(n)*phi(n).

Links

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

Crossrefs

Cf. A000005 (tau), A000010 (phi), A008683 (mu), A359954.

Programs

Mathematica

f[p_, e_] := If[e == 1, 2*p^2 - 2*p - 1, ((p^2 - 1)*e + p^2)*(p - 1)*p^(2*e - 3)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 16 2023 *)

PARI

a(n) = sumdiv(n, d, d*numdiv(d)*eulerphi(d)*moebius(n/d)); \\ Michel Marcus, May 13 2023

Formula

Multiplicative with a(p) = 2*p^2 - 2*p - 1 and a(p^e) = ((p^2 - 1)*e + p^2)*(p - 1)*p^(2*e - 3) for e > 1. - Amiram Eldar, May 16 2023