cp's OEIS Frontend

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

%I A190117 #20 Jun 22 2025 07:29:50
%S A190117 0,2,5,21,26,56,63,159,213,283,294,486,499,625,745,1257,1274,1652,
%T A190117 1671,2151,2361,2647,2670,3726,3976,4366,5095,5991,6020,6950,6981,
%U A190117 9541,10003,10649,11069,13229,13266,14064,14688,17408,17449,19171,19214,21326,23081,24231,24278,29654,30340,32590
%N A190117 a(n) = Sum_{k=1..n} k*k', where n' is the arithmetic derivative of n.
%H A190117 G. C. Greubel, <a href="/A190117/b190117.txt">Table of n, a(n) for n = 1..10000</a>
%F A190117 a(n) ~ c * n^3 / 3, where c = Sum_{p prime} 1/(p*(p-1)) = A136141. - _Amiram Eldar_, Jun 22 2025
%e A190117 1*1' + 2*2' + 3*3' = 0 + 2 + 3 = 5 -> a(3) = 5.
%p A190117 der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]):
%p A190117 seq(add(der(i)*i,i=1..n),n=1..50);
%t A190117 A003415[n_]:= If[Abs@n < 2, 0, n Total[#2/#1 & @@@FactorInteger[Abs@n]]];
%t A190117 Table[Sum[k*A003415[k], {k, 1, n}], {n, 1, 50}] (* _G. C. Greubel_, Dec 29 2017 *)
%Y A190117 Partial sums of A190116.
%Y A190117 Cf. A003415, A136141.
%K A190117 nonn
%O A190117 1,2
%A A190117 _Giorgio Balzarotti_, May 04 2011