A332619 a(n) = Sum_{d|n} lcm(d, n/d) / d.

1, 3, 4, 6, 6, 12, 8, 12, 11, 18, 12, 24, 14, 24, 24, 23, 18, 33, 20, 36, 32, 36, 24, 48, 27, 42, 32, 48, 30, 72, 32, 45, 48, 54, 48, 66, 38, 60, 56, 72, 42, 96, 44, 72, 66, 72, 48, 92, 51, 81, 72, 84, 54, 96, 72, 96, 80, 90, 60, 144, 62, 96, 88, 88, 84, 144, 68, 108, 96, 144

Offset

1,2

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.

Formula

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

From Amiram Eldar, Dec 05 2022: (Start)

Multiplicative with a(p^e) = (p^(e+2)-1)/(p^2-1) + e/2 if e is even, and (p^(e+2)-p)/(p^2-1) + (e + 1)/2 if e is odd.

Sum_{k=1..n} a(k) ~ c * n^2, where c = 7*zeta(6)/(8*zeta(5)) = 0.740543... . (End)

Maple

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

Mathematica

Table[Sum[LCM[d, n/d]/d, {d, Divisors[n]}], {n, 1, 70}]
f[p_, e_] := If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1) + e/2, (p^(e + 2) - p)/(p^2 - 1) + (e + 1)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

Prog

(PARI) A332619(n) = sumdiv(n, d, lcm(d, n/d)/d); \\ Antti Karttunen, Nov 12 2021

Crossrefs

Cf. A055155, A057660, A057661, A057670, A332618.

Cf. A013663, A013664.

Sequence in context: A230593 A304411 A360522 * A322319 A001615 A158523

Adjacent sequences: A332616 A332617 A332618 * A332620 A332621 A332622

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Feb 17 2020

Status

approved