A190119 - OEIS

%I #22 Jul 25 2024 06:58:22

%S 0,1,2,3,4,9,10,13,15,22,23,27,28,37,45,47,48,55,56,62,72,85,86,97,99,

%T 114,115,123,124,155,156,161,175,194,206,211,212,233,249,266,267,308,

%U 309,321,334,359,360,367,369,378,398,412,413,416,432,455,477,508,509,532,533,566,583,586,604,665,666,684,710,769

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

%C Use lcm(1,0)=0.

%H G. C. Greubel, <a href="/A190119/b190119.txt">Table of n, a(n) for n = 1..10000</a>

%e lcm(1,1')/1+lcm(2,2')/2+lcm(3,3')/3=0+2/2+3/3=2 ->a(3)=2.

%p der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]):

%p seq(add(lcm(der(i),i)/i,i=1..n),n=1..50);

%t A003415[n_]:= If[Abs@n < 2, 0, n Total[#2/#1 & @@@FactorInteger[Abs@n]]]; Table[Sum[LCM[k, A003415[k]]/k, {k, 1, n}], {n,1,50}] (* _G. C. Greubel_, Dec 29 2017 *)

%o (PARI) {A003415(n, f)=sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i])};

%o for(n=1,20, print1(sum(k=1,n,lcm(k,A003415(k))/k), ", ")) \\ _G. C. Greubel_, Dec 29 2017

%Y Cf. A003415.

%K nonn

%O 1,3

%A _Giorgio Balzarotti_, May 04 2011