A062952 Multiplicative with a(p^e) = (p^(2*e+2)-p^(e+1)-p^e+p)/(p^2-1).

1, 4, 9, 18, 25, 36, 49, 78, 87, 100, 121, 162, 169, 196, 225, 326, 289, 348, 361, 450, 441, 484, 529, 702, 645, 676, 807, 882, 841, 900, 961, 1334, 1089, 1156, 1225, 1566, 1369, 1444, 1521, 1950, 1681, 1764, 1849, 2178, 2175, 2116, 2209, 2934, 2443, 2580

Offset: 1

Author

Vladeta Jovovic, Jul 21 2001

Keywords

• nonn
• mult

Comments

If k is squarefree (cf. A005117) then A062952(k) = k^2. - Benoit Cloitre, Apr 16 2002

Inverse Möbius transform of A062354(n). - Wesley Ivan Hurt, Jul 26 2025

Links

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

Crossrefs

Cf. A005117, A029939, A057660, A062952.

Cf. A002117, A013661, A183699, A330523.

Cf. A062354.

Programs

Mathematica

f[p_, e_] := (p^(2*e+2)-p^(e+1)-p^e+p)/(p^2-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 50] (* Amiram Eldar, Jul 31 2019 *)

PARI

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)) \\ Michel Marcus, Jun 17 2013

Formula

a(n) = Sum_{d|n} phi(d)*sigma(d).

a(n) = Sum_{k=1..n} sigma(n/gcd(n, k)).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A183699 * A330523 / 3. - Amiram Eldar, Oct 30 2022