A332618 a(n) = Sum_{d|n} lcm(d, n/d) / gcd(d, n/d).

1, 4, 6, 9, 10, 24, 14, 20, 19, 40, 22, 54, 26, 56, 60, 41, 34, 76, 38, 90, 84, 88, 46, 120, 51, 104, 60, 126, 58, 240, 62, 84, 132, 136, 140, 171, 74, 152, 156, 200, 82, 336, 86, 198, 190, 184, 94, 246, 99, 204, 204, 234, 106, 240, 220, 280, 228, 232, 118, 540

Offset: 1

Author

Ilya Gutkovskiy, Feb 17 2020

Keywords

• nonn
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences related to gcd's.
• Index entries for sequences related to lcm's.

Crossrefs

Cf. A007913, A055155, A056789, A057670, A332619, A337180.

Programs

Maple

a:= n-> n*add(1/igcd(d, n/d)^2, d=numtheory[divisors](n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 17 2020

Mathematica

Table[Sum[LCM[d, n/d]/GCD[d, n/d], {d, Divisors[n]}], {n, 1, 60}]
f[p_, e_] := If[EvenQ[e], (2*p^(e+2) - p^2 - 1)/(p^2 - 1), 2*(p^(e+2) - p)/(p^2 - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

PARI

A332618(n) = sumdiv(n,d,lcm(d,n/d)/gcd(d,n/d)); \\ Antti Karttunen, Nov 12 2021

Formula

a(n) = n * Sum_{d|n} 1 / gcd(d, n/d)^2.

Multiplicative with a(p^e) = (2*p^(e+2) - p^2 - 1)/(p^2 - 1) if e is even, a(p^e) = 2*(p^(e+2) - p)/(p^2 - 1) if e is odd. - Sebastian Karlsson, May 07 2022

From Peter Bala, Jan 24 2024: (Start)

a(n) = Sum_{d divides n} A007913(d)*n/d.

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