cp's OEIS Frontend

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

%I A006580 M3336 #49 Apr 28 2023 08:16:56
%S A006580 0,0,1,4,8,20,21,56,60,96,105,220,152,364,301,360,464,816,549,1140,
%T A006580 760,1036,1221,2024,1196,2200,2041,2484,2184,4060,2205,4960,3664,4224,
%U A006580 4641,5180,4008,8436,6517,7072,5980,11480,6321,13244,8888,9540,11661,17296
%N A006580 a(n) = Sum_{k=1..n-1} lcm(k,n-k).
%D A006580 _Marc LeBrun_, personal communication.
%D A006580 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006580 Alois P. Heinz, <a href="/A006580/b006580.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms from Reinhard Zumkeller)
%H A006580 Marc Le Brun, <a href="/A006577/a006577.pdf">Email to N. J. A. Sloane, Jul 1991</a>.
%F A006580 For n > 0, a(n) = (n/6)*Sum_{d|n} (d*phi(d) - A023900(d)). - _Sebastian Karlsson_, Oct 02 2021
%F A006580 a(n) = (n/6) * (A057660(n) - A130054(n)), for n > 0. - _Amiram Eldar_, Apr 28 2023
%p A006580 a:= n-> add(ilcm(j, n-j), j=0..n):
%p A006580 seq(a(n), n=0..70);  # _Alois P. Heinz_, Aug 25 2019
%t A006580 Table[ Sum[ LCM[ k, n-k ], {k, 1, n-1} ], {n, 2, 50} ] (* _Olivier Gérard_, Aug 15 1997 *)
%t A006580 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 *)
%o A006580 (Haskell)
%o A006580 a006580 n = a006580_list !! (n-1)
%o A006580 a006580_list = map sum a003990_tabl
%o A006580 -- _Reinhard Zumkeller_, Aug 05 2012
%o A006580 (PARI) a(n) = sum(k=1, n-1, lcm(k, n-k)); \\ _Michel Marcus_, Aug 11 2017
%Y A006580 Antidiagonal sums of array A003990.
%Y A006580 Cf. A209295.
%Y A006580 Cf. A000010, A023900, A057660, A130054.
%K A006580 nonn
%O A006580 0,4
%A A006580 _N. J. A. Sloane_
%E A006580 More terms from _Olivier Gérard_, Aug 15 1997