A190117 a(n) = Sum_{k=1..n} k*k', where n' is the arithmetic derivative of n.

0, 2, 5, 21, 26, 56, 63, 159, 213, 283, 294, 486, 499, 625, 745, 1257, 1274, 1652, 1671, 2151, 2361, 2647, 2670, 3726, 3976, 4366, 5095, 5991, 6020, 6950, 6981, 9541, 10003, 10649, 11069, 13229, 13266, 14064, 14688, 17408, 17449, 19171, 19214, 21326, 23081, 24231, 24278, 29654, 30340, 32590

Offset: 1

Author

Giorgio Balzarotti, May 04 2011

Keywords

• nonn

Examples

			1*1' + 2*2' + 3*3' = 0 + 2 + 3 = 5 -> a(3) = 5.
		

Links

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

Crossrefs

Partial sums of A190116.

Cf. A003415, A136141.

Programs

Maple

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

Mathematica

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

Formula

a(n) ~ c * n^3 / 3, where c = Sum_{p prime} 1/(p*(p-1)) = A136141. - Amiram Eldar, Jun 22 2025