A006580 a(n) = Sum_{k=1..n-1} lcm(k,n-k). (Formerly M3336)

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

0,4

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.

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

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 *)

Prog

(Haskell)
a006580 n = a006580_list !! (n-1)
a006580_list = map sum a003990_tabl
-- Reinhard Zumkeller, Aug 05 2012
(PARI) a(n) = sum(k=1, n-1, lcm(k, n-k)); \\ Michel Marcus, Aug 11 2017

Crossrefs

Antidiagonal sums of array A003990.

Cf. A209295.

Cf. A000010, A023900, A057660, A130054.

Sequence in context: A273143 A273174 A178447 * A061814 A087254 A160726

Adjacent sequences: A006577 A006578 A006579 * A006581 A006582 A006583

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Olivier Gérard, Aug 15 1997

Status

approved