A190119 -id:A190119 - cp's OEIS frontend

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

0, 2, 5, 6, 11, 41, 48, 54, 60, 130, 141, 153, 166, 292, 412, 414, 431, 473, 492, 522, 732, 1018, 1041, 1107, 1117, 1507, 1508, 1564, 1593, 2523, 2554, 2564, 3026, 3672, 4092, 4107, 4144, 4942, 5566, 5736, 5777, 7499, 7542, 7674, 7869, 9019, 9066, 9087, 9101, 9191

Offset: 1

Giorgio Balzarotti, May 04 2011

nonn

Use lcm(1,0)=0 and gcd(1,0)=1.

			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.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A003415, A190118, A190119.

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]):
seq(add(lcm(der(i),i)/gcd(der(i),i),i=1..n),n=1..50);

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

{A003415(n, f)=sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i])};
for(n=1, 50, print1(sum(k=1,n,lcm(k,A003415(k))/gcd(k,A003145(k))), ", ")) \\ G. C. Greubel, Dec 29 2017

A190122 a(n) = Sum_{k=1..n} k*lcm(k,k')/gcd(k,k'), where k' is arithmetic derivative of k.

Original entry on oeis.org

0, 4, 13, 17, 42, 222, 271, 319, 373, 1073, 1194, 1338, 1507, 3271, 5071, 5103, 5392, 6148, 6509, 7109, 11519, 17811, 18340, 19924, 20174, 30314, 30341, 31909, 32750, 60650, 61611, 61931, 77177, 99141, 113841, 114381, 115750, 146074, 170410, 177210, 178891

Offset: 1

Giorgio Balzarotti, May 04 2011

nonn

Use lcm(1,0)=0 and gcd(1,0)=1.

			lcm(1,1')/gcd(1,1')*1+lcm(2,2')/gcd(2,2')*2+lcm(3,3')/gcd(3,3')*3=0+2/1*2+3/1*3=13 ->a(3)=13.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A003415, A189036, A190118, A190119, A190120.

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]):
seq(add(lcm(der(i),i)/gcd(der(i),i)*i,i=1..n),n=1..50);

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

a(n) = Sum_{k=1..n} k*A189036(k).

cp's OEIS Frontend

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

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