%I #12 Nov 21 2013 05:32:22
%S 0,0,0,4,4,5,5,21,26,27,27,59,59,65,77,157,157,167,167,211,218,219,
%T 219,267,274,282,309,389,389,390,390,566,575,576,592,684,684,694,726,
%U 798,798,799,799,911,927,937,937,1177,1186,1225,1249,1341,1341,1449,1481
%N Partial sums of second arithmetic derivative of natural numbers.
%C a(n) = 1''+2''+3''+4''+5''+...+n'' -> ~ constant * n^2 as n -> oo.
%C Note: a(n) = sum(D^d(i)^m,i=1..n) -> constant * n^(m+1) as n -> oo where D^d(i) is the derivative of order d th of the natural number i (results on arithmetic derivatives descent from Barbeau's paper in References).
%H E. J. Barbeau, <a href="http://cms.math.ca/cmb/v4/p117">Remark on an arithmetic derivative</a>, Canad. Math. Bull. vol. 4, no. 2, May 1961.
%F a(n) = sum(i'', i=1..n), where i'' is the second arithmetic derivative of i (A068346).
%e a(5) = 1'' + 2'' + 3'' + 4'' + 5'' = 0+0+0+4+0 = 4.
%p der:=n->n*add(op(2, p)/op(1, p), p=ifactors(n)[2]): seq(add(der(der(i)),i=1..j),j=1..55);
%Y Cf. A003415, A068346, A190121, A231946.
%K nonn
%O 1,4
%A _Giorgio Balzarotti_, Nov 19 2013