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A190120 a(n) = Sum_{k=1..n} lcm(k,k')/gcd(k,k'), where n' is arithmetic derivative of n.

%I A190120 #22 Jul 25 2024 06:58:18
%S A190120 0,2,5,6,11,41,48,54,60,130,141,153,166,292,412,414,431,473,492,522,
%T A190120 732,1018,1041,1107,1117,1507,1508,1564,1593,2523,2554,2564,3026,3672,
%U A190120 4092,4107,4144,4942,5566,5736,5777,7499,7542,7674,7869,9019,9066,9087,9101,9191
%N A190120 a(n) = Sum_{k=1..n} lcm(k,k')/gcd(k,k'), where n' is arithmetic derivative of n.
%C A190120 Use lcm(1,0)=0 and gcd(1,0)=1.
%H A190120 G. C. Greubel, <a href="/A190120/b190120.txt">Table of n, a(n) for n = 1..10000</a>
%e A190120 lcm(1,1')/gcd(1,1')+lcm(2,2')/gcd(2,2')+lcm(3,3')/gcd(3,3')=0+2/1+3/1=5 ->a(3)=5.
%p A190120 der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]):
%p A190120 seq(add(lcm(der(i),i)/gcd(der(i),i),i=1..n),n=1..50);
%t A190120 A003415[n_]:= If[Abs@n < 2, 0, n Total[#2/#1 & @@@FactorInteger[Abs@n]]];
%t A190120 Table[Sum[LCM[k, A003415[k]]/GCD[k, A003415[k]], {k, 1, n}], {n,1,50}] (* _G. C. Greubel_, Dec 29 2017 *)
%o A190120 (PARI) {A003415(n, f)=sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i])};
%o A190120 for(n=1, 50, print1(sum(k=1,n,lcm(k,A003415(k))/gcd(k,A003145(k))), ", ")) \\ _G. C. Greubel_, Dec 29 2017
%Y A190120 Cf. A003415, A190118, A190119.
%K A190120 nonn
%O A190120 1,2
%A A190120 _Giorgio Balzarotti_, May 04 2011