A006580 a(n) = Sum_{k=1..n-1} lcm(k,n-k).

0, 0, 1, 4, 8, 20, 21, 56, 60, 96, 105, 220, 152, 364, 301, 360, 464, 816, 549, 1140, 760, 1036, 1221, 2024, 1196, 2200, 2041, 2484, 2184, 4060, 2205, 4960, 3664, 4224, 4641, 5180, 4008, 8436, 6517, 7072, 5980, 11480, 6321, 13244, 8888, 9540, 11661, 17296

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• Marc LeBrun, personal communication.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1000 terms from Reinhard Zumkeller)
• Marc Le Brun, Email to N. J. A. Sloane, Jul 1991.

Crossrefs

Antidiagonal sums of array A003990.

Cf. A209295.

Cf. A000010, A023900, A057660, A130054.

Programs

Haskell

a006580 n = a006580_list !! (n-1)
a006580_list = map sum a003990_tabl
-- Reinhard Zumkeller, Aug 05 2012

Maple

a:= n-> add(ilcm(j, n-j), j=0..n):
seq(a(n), n=0..70);  # Alois P. Heinz, Aug 25 2019

Mathematica

Table[ Sum[ LCM[ k, n-k ], {k, 1, n-1} ], {n, 2, 50} ] (* Olivier Gérard, Aug 15 1997 *)
f1[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); f2[p_, e_] := 1 - (p - 1)*e; a[n_] := (Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct)*n/6; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Apr 28 2023 *)

PARI

a(n) = sum(k=1, n-1, lcm(k, n-k)); \\ Michel Marcus, Aug 11 2017

Formula

For n > 0, a(n) = (n/6)*Sum_{d|n} (d*phi(d) - A023900(d)). - Sebastian Karlsson, Oct 02 2021

a(n) = (n/6) * (A057660(n) - A130054(n)), for n > 0. - Amiram Eldar, Apr 28 2023

Extensions

More terms from Olivier Gérard, Aug 15 1997