|
%I #48 Apr 26 2023 06:11:47
%S 1,4,12,24,55,66,154,176,279,320,616,468,1027,910,1110,1376,2329,1656,
%T 3268,2320,3171,3674,5842,3624,6525,6136,7398,6636,11803,6630,14446,
%U 10944,12837,13940,15820,12096,24679,19570,21450,18080,33661,18984,38872,26884
%N a(n) = Sum_{k=1..n} lcm(n,k).
%H Akshay Bansal, <a href="/A051193/b051193.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H Akshay Bansal, <a href="/A051193/a051193.c.txt">C Program</a>.
%H Soichi Ikeda and Kaneaki Matsuoka, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Ikeda/ikeda4.html">On the Lcm-Sum Function</a>, Journal of Integer Sequences, Vol. 17 (2014), Article 14.1.7.
%H László Tóth, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Toth/toth9.html">Weighted gcd-sum functions</a>, J. Integer Sequences, 14 (2011), Article 11.7.7.
%H <a href="/index/Lc#lcm">Index entries for sequences related to lcm's</a>.
%F a(n) = n*(1+Sum_{d|n} d*phi(d))/2 = n*(1+A057660(n))/2 = n*A057661(n). - _Vladeta Jovovic_, Jun 21 2002
%F G.f.: x*f'(x), where f(x) = x/(2*(1 - x)) + (1/2)*Sum_{k>=1} k*phi(k)*x^k/(1 - x^k) and phi() is the Euler totient function (A000010). - _Ilya Gutkovskiy_, Aug 31 2017
%F Sum_{k=1..n} a(k) ~ 3 * zeta(3) * n^4 / (4*Pi^2). - _Vaclav Kotesovec_, May 29 2021
%p a:=n->add(ilcm( n, j ), j=1..n): seq(a(n), n=1..50); # _Zerinvary Lajos_, Nov 07 2006
%t Table[Sum[LCM[k, n], {k, 1, n}], {n, 1, 39}] (* _Geoffrey Critzer_, Feb 16 2015 *)
%t f[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); a[n_] := n * (1 + Times @@ f @@@ FactorInteger[n])/2; Array[a, 100] (* _Amiram Eldar_, Apr 26 2023 *)
%o (Haskell)
%o a051193 = sum . a051173_row -- _Reinhard Zumkeller_, Feb 11 2014
%o (PARI) a(n) = sum(k=1, n, lcm(n,k)); \\ _Michel Marcus_, Feb 06 2015
%Y Cf. A000010, A018804, A051173 (triangle whose n-th row sum is a(n)), A057660, A057661.
%K nonn
%O 1,2
%A _Clark Kimberling_
|
